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Automated Planet Finder

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The Automated Planet Finder ( APF ) Telescope a.k.a. Rocky Planet Finder , is a fully robotic 2.4-meter optical telescope at Lick Observatory , situated on the summit of Mount Hamilton, east of San Jose, California, USA. It is designed to search for extrasolar planets in the range of five to twenty times the mass of the Earth . The instrument will examine about 10 stars per night. Over the span of a decade, the telescope is expected to study 1,000 nearby stars for planets. Its estimated cost was $ 10 million. The total cost-to-completion of the APF project was $ 12.37 million. First light was originally scheduled for 2006, but delays in the construction of the major components of the telescope pushed this back to August 2013. It was commissioned in August 2013.

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85-529: The telescope uses high-precision radial velocity measurements to measure the gravitational reflex motion of nearby stars caused by the orbiting of planets. The design goal is to detect stellar motions as small as one meter per second, comparable to a slow walking speed. The main targets will be stars within about 100 light years of the Earth. Early tests show that the performance of the Ken and Gloria Levy Doppler Spectrometer

170-415: A . {\displaystyle (f\circ g)'(a)=\lim _{x\to a}{\frac {f(g(x))-f(g(a))}{x-a}}.} Assume for the moment that g ( x ) {\displaystyle g(x)\!} does not equal g ( a ) {\displaystyle g(a)} for any x {\displaystyle x} near a {\displaystyle a} . Then the previous expression

255-949: A . . b ( x ) = x {\displaystyle f_{a\,.\,.\,b}(x)=x} when b < a {\displaystyle b<a} . Then the chain rule takes the form D f 1 . . n = ( D f 1 ∘ f 2 . . n ) ( D f 2 ∘ f 3 . . n ) ⋯ ( D f n − 1 ∘ f n . . n ) D f n = ∏ k = 1 n [ D f k ∘ f ( k + 1 ) . . n ] {\displaystyle Df_{1\,.\,.\,n}=(Df_{1}\circ f_{2\,.\,.\,n})(Df_{2}\circ f_{3\,.\,.\,n})\cdots (Df_{n-1}\circ f_{n\,.\,.\,n})Df_{n}=\prod _{k=1}^{n}\left[Df_{k}\circ f_{(k+1)\,.\,.\,n}\right]} or, in

340-445: A . . b = f a ∘ f a + 1 ∘ ⋯ ∘ f b − 1 ∘ f b {\displaystyle f_{a\,.\,.\,b}=f_{a}\circ f_{a+1}\circ \cdots \circ f_{b-1}\circ f_{b}} where f a . . a = f a {\displaystyle f_{a\,.\,.\,a}=f_{a}} and f

425-716: A ) ) ⋅ d u d v | v = h ( a ) ⋅ d v d x | x = a , {\displaystyle {\frac {dy}{dx}}=\left.{\frac {dy}{du}}\right|_{u=g(h(a))}\cdot \left.{\frac {du}{dv}}\right|_{v=h(a)}\cdot \left.{\frac {dv}{dx}}\right|_{x=a},} or for short, d y d x = d y d u ⋅ d u d v ⋅ d v d x . {\displaystyle {\frac {dy}{dx}}={\frac {dy}{du}}\cdot {\frac {du}{dv}}\cdot {\frac {dv}{dx}}.} The derivative function

510-439: A ) ) {\displaystyle f(g(x))-f(g(a))=q(g(x))(g(x)-g(a))} and g ( x ) − g ( a ) = r ( x ) ( x − a ) . {\displaystyle g(x)-g(a)=r(x)(x-a).} Therefore, f ( g ( x ) ) − f ( g ( a ) ) = q ( g ( x ) ) r ( x ) ( x −

595-407: A ) ) k h + η ( k h ) k h . {\displaystyle f(g(a)+k_{h})-f(g(a))=f'(g(a))k_{h}+\eta (k_{h})k_{h}.} To study the behavior of this expression as h tends to zero, expand k h . After regrouping the terms, the right-hand side becomes: f ′ ( g ( a ) ) g ′ (

680-768: A ) ) ⋅ g ′ ( h ( a ) ) ⋅ h ′ ( a ) = ( f ′ ∘ g ∘ h ) ( a ) ⋅ ( g ′ ∘ h ) ( a ) ⋅ h ′ ( a ) . {\displaystyle {\begin{aligned}(f\circ g\circ h)'(a)&=f'((g\circ h)(a))\cdot (g\circ h)'(a)\\&=f'((g\circ h)(a))\cdot g'(h(a))\cdot h'(a)\\&=(f'\circ g\circ h)(a)\cdot (g'\circ h)(a)\cdot h'(a).\end{aligned}}} In Leibniz's notation , this is: d y d x = d y d u | u = g ( h (

765-683: A ) ) ⋅ h ′ ( a ) . {\displaystyle (f\circ g\circ h)'(a)=(f\circ g)'(h(a))\cdot h'(a)=f'(g(h(a)))\cdot g'(h(a))\cdot h'(a).} This is the same as what was computed above. This should be expected because ( f ∘ g ) ∘ h = f ∘ ( g ∘ h ) . Sometimes, it is necessary to differentiate an arbitrarily long composition of the form f 1 ∘ f 2 ∘ ⋯ ∘ f n − 1 ∘ f n {\displaystyle f_{1}\circ f_{2}\circ \cdots \circ f_{n-1}\circ f_{n}\!} . In this case, define f

850-401: A ) ) = f ( g ( a ) + g ′ ( a ) h + ε ( h ) h ) − f ( g ( a ) ) . {\displaystyle f(g(a+h))-f(g(a))=f(g(a)+g'(a)h+\varepsilon (h)h)-f(g(a)).} The next step is to use the definition of differentiability of f at g ( a ). This requires a term of the form f ( g (

935-610: A ) , {\displaystyle f(g(x))-f(g(a))=q(g(x))r(x)(x-a),} but the function given by h ( x ) = q ( g ( x )) r ( x ) is continuous at a , and we get, for this a ( f ( g ( a ) ) ) ′ = q ( g ( a ) ) r ( a ) = f ′ ( g ( a ) ) g ′ ( a ) . {\displaystyle (f(g(a)))'=q(g(a))r(a)=f'(g(a))g'(a).} A similar approach works for continuously differentiable (vector-)functions of many variables. This method of factoring also allows

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1020-411: A ) h + ε ( h ) h . {\displaystyle g(a+h)-g(a)=g'(a)h+\varepsilon (h)h.} Here the left-hand side represents the true difference between the value of g at a and at a + h , whereas the right-hand side represents the approximation determined by the derivative plus an error term. In the situation of the chain rule, such a function ε exists because g

1105-463: A ) h + [ f ′ ( g ( a ) ) ε ( h ) + η ( k h ) g ′ ( a ) + η ( k h ) ε ( h ) ] h . {\displaystyle f'(g(a))g'(a)h+[f'(g(a))\varepsilon (h)+\eta (k_{h})g'(a)+\eta (k_{h})\varepsilon (h)]h.} Because ε ( h ) and η ( k h ) tend to zero as h tends to zero,

1190-416: A exists and equals Q ( g ( a )) , which is f ′( g ( a )) . This shows that the limits of both factors exist and that they equal f ′( g ( a )) and g ′( a ) , respectively. Therefore, the derivative of f ∘ g at a exists and equals f ′( g ( a )) g ′( a ) . Another way of proving the chain rule is to measure the error in the linear approximation determined by the derivative. This proof has

1275-397: A exists and equals g ′( a ) . As for Q ( g ( x )) , notice that Q is defined wherever f is. Furthermore, f is differentiable at g ( a ) by assumption, so Q is continuous at g ( a ) , by definition of the derivative. The function g is continuous at a because it is differentiable at a , and therefore Q ∘ g is continuous at a . So its limit as x goes to

1360-410: A ) + k ) for some k . In the above equation, the correct k varies with h . Set k h = g ′( a ) h + ε ( h ) h and the right hand side becomes f ( g ( a ) + k h ) − f ( g ( a )) . Applying the definition of the derivative gives: f ( g ( a ) + k h ) − f ( g ( a ) ) = f ′ ( g (

1445-474: A ) , then the difference quotient for f ∘ g is zero because f ( g ( x )) equals f ( g ( a )) , and the above product is zero because it equals f ′( g ( a )) times zero. So the above product is always equal to the difference quotient, and to show that the derivative of f ∘ g at a exists and to determine its value, we need only show that the limit as x goes to a of the above product exists and determine its value. To do this, recall that

1530-421: A , then it might happen that no matter how close one gets to a , there is always an even closer x such that g ( x ) = g ( a ) . For example, this happens near a = 0 for the continuous function g defined by g ( x ) = 0 for x = 0 and g ( x ) = x sin(1/ x ) otherwise. Whenever this happens, the above expression is undefined because it involves division by zero . To work around this, introduce

1615-416: A 1676 memoir (with a sign error in the calculation). The common notation of the chain rule is due to Leibniz. Guillaume de l'Hôpital used the chain rule implicitly in his Analyse des infiniment petits . The chain rule does not appear in any of Leonhard Euler 's analysis books, even though they were written over a hundred years after Leibniz's discovery. . It is believed that the first "modern" version of

1700-406: A car travels twice as fast as a bicycle and the bicycle is four times as fast as a walking man, then the car travels 2 × 4 = 8 times as fast as the man." The relationship between this example and the chain rule is as follows. Let z , y and x be the (variable) positions of the car, the bicycle, and the walking man, respectively. The rate of change of relative positions of the car and the bicycle

1785-575: A function Q {\displaystyle Q} as follows: Q ( y ) = { f ( y ) − f ( g ( a ) ) y − g ( a ) , y ≠ g ( a ) , f ′ ( g ( a ) ) , y = g ( a ) . {\displaystyle Q(y)={\begin{cases}\displaystyle {\frac {f(y)-f(g(a))}{y-g(a)}},&y\neq g(a),\\f'(g(a)),&y=g(a).\end{cases}}} We will show that

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1870-734: A function of an independent variable y , we substitute f ( y ) {\displaystyle f(y)} for x wherever it appears. Then we can solve for f' . f ′ ( g ( f ( y ) ) ) g ′ ( f ( y ) ) = 1 f ′ ( y ) g ′ ( f ( y ) ) = 1 f ′ ( y ) = 1 g ′ ( f ( y ) ) . {\displaystyle {\begin{aligned}f'(g(f(y)))g'(f(y))&=1\\f'(y)g'(f(y))&=1\\f'(y)={\frac {1}{g'(f(y))}}.\end{aligned}}} For example, consider

1955-433: A high-resolution spectrum and comparing the measured wavelengths of known spectral lines to wavelengths from laboratory measurements. A positive radial velocity indicates the distance between the objects is or was increasing; a negative radial velocity indicates the distance between the source and observer is or was decreasing. William Huggins ventured in 1868 to estimate the radial velocity of Sirius with respect to

2040-457: A much smaller planet with an orbital plane on the line of sight. It has been suggested that planets with high eccentricities calculated by this method may in fact be two-planet systems of circular or near-circular resonant orbit. The radial velocity method to detect exoplanets is based on the detection of variations in the velocity of the central star, due to the changing direction of the gravitational pull from an (unseen) exoplanet as it orbits

2125-612: A unified approach to stronger forms of differentiability, when the derivative is required to be Lipschitz continuous , Hölder continuous , etc. Differentiation itself can be viewed as the polynomial remainder theorem (the little Bézout theorem, or factor theorem), generalized to an appropriate class of functions. If y = f ( x ) {\displaystyle y=f(x)} and x = g ( t ) {\displaystyle x=g(t)} then choosing infinitesimal Δ t ≠ 0 {\displaystyle \Delta t\not =0} we compute

2210-882: A variable z depends on the variable y , which itself depends on the variable x (that is, y and z are dependent variables ), then z depends on x as well, via the intermediate variable y . In this case, the chain rule is expressed as d z d x = d z d y ⋅ d y d x , {\displaystyle {\frac {dz}{dx}}={\frac {dz}{dy}}\cdot {\frac {dy}{dx}},} and d z d x | x = d z d y | y ( x ) ⋅ d y d x | x , {\displaystyle \left.{\frac {dz}{dx}}\right|_{x}=\left.{\frac {dz}{dy}}\right|_{y(x)}\cdot \left.{\frac {dy}{dx}}\right|_{x},} for indicating at which points

2295-607: Is d z d y = 2. {\textstyle {\frac {dz}{dy}}=2.} Similarly, d y d x = 4. {\textstyle {\frac {dy}{dx}}=4.} So, the rate of change of the relative positions of the car and the walking man is d z d x = d z d y ⋅ d y d x = 2 ⋅ 4 = 8. {\displaystyle {\frac {dz}{dx}}={\frac {dz}{dy}}\cdot {\frac {dy}{dx}}=2\cdot 4=8.} The rate of change of positions

2380-458: Is a function that is differentiable at g ( c ) , then the composite function f ∘ g {\displaystyle f\circ g} is differentiable at c , and the derivative is ( f ∘ g ) ′ ( c ) = f ′ ( g ( c ) ) ⋅ g ′ ( c ) . {\displaystyle (f\circ g)'(c)=f'(g(c))\cdot g'(c).} The rule

2465-470: Is also an application of the chain rule. The chain rule seems to have first been used by Gottfried Wilhelm Leibniz . He used it to calculate the derivative of a + b z + c z 2 {\displaystyle {\sqrt {a+bz+cz^{2}}}} as the composite of the square root function and the function a + b z + c z 2 {\displaystyle a+bz+cz^{2}\!} . He first mentioned it in

2550-411: Is also differentiable. This formula can fail when one of these conditions is not true. For example, consider g ( x ) = x . Its inverse is f ( y ) = y , which is not differentiable at zero. If we attempt to use the above formula to compute the derivative of f at zero, then we must evaluate 1/ g ′( f (0)) . Since f (0) = 0 and g ′(0) = 0 , we must evaluate 1/0, which is undefined. Therefore,

2635-423: Is assumed that η ( k ) tends to zero as k tends to zero. If we set η (0) = 0 , then η is continuous at 0. Proving the theorem requires studying the difference f ( g ( a + h )) − f ( g ( a )) as h tends to zero. The first step is to substitute for g ( a + h ) using the definition of differentiability of g at a : f ( g ( a + h ) ) − f ( g (

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2720-491: Is assumed to be differentiable at a . Again by assumption, a similar function also exists for f at g ( a ). Calling this function η , we have f ( g ( a ) + k ) − f ( g ( a ) ) = f ′ ( g ( a ) ) k + η ( k ) k . {\displaystyle f(g(a)+k)-f(g(a))=f'(g(a))k+\eta (k)k.} The above definition imposes no constraints on η (0), even though it

2805-527: Is determined by astrometric observations (for example, a secular change in the annual parallax ). Light from an object with a substantial relative radial velocity at emission will be subject to the Doppler effect , so the frequency of the light decreases for objects that were receding ( redshift ) and increases for objects that were approaching ( blueshift ). The radial velocity of a star or other luminous distant objects can be measured accurately by taking

2890-744: Is differentiable at its immediate input, then the composite function is also differentiable by the repeated application of Chain Rule, where the derivative is (in Leibniz's notation): d f 1 d x = d f 1 d f 2 d f 2 d f 3 ⋯ d f n d x . {\displaystyle {\frac {df_{1}}{dx}}={\frac {df_{1}}{df_{2}}}{\frac {df_{2}}{df_{3}}}\cdots {\frac {df_{n}}{dx}}.} The chain rule can be applied to composites of more than two functions. To take

2975-526: Is equal to the product of two factors: lim x → a f ( g ( x ) ) − f ( g ( a ) ) g ( x ) − g ( a ) ⋅ g ( x ) − g ( a ) x − a . {\displaystyle \lim _{x\to a}{\frac {f(g(x))-f(g(a))}{g(x)-g(a)}}\cdot {\frac {g(x)-g(a)}{x-a}}.} If g {\displaystyle g} oscillates near

3060-478: Is meeting the design goals. The spectrometer has high throughput and is meeting the design sensitivity of (1.0 m/s), similar to the radial velocity precision of HARPS and HIRES . Parts for the telescopes were constructed by international companies: The telescope is also being used to search for optical signals coming from laser transmissions from hypothetical extraterrestrial civilizations ( search for extraterrestrial intelligence - SETI). This undertaking

3145-526: Is performed for the heavily funded Breakthrough Listen project of the Berkeley SETI Research Center . Radial velocity The radial velocity or line-of-sight velocity of a target with respect to an observer is the rate of change of the vector displacement between the two points. It is formulated as the vector projection of the target-observer relative velocity onto the relative direction or line-of-sight (LOS) connecting

3230-686: Is rather technical. However, it is simpler to write in the case of functions of the form f ( g 1 ( x ) , … , g k ( x ) ) , {\displaystyle f(g_{1}(x),\dots ,g_{k}(x)),} where f : R k → R {\displaystyle f:\mathbb {R} ^{k}\to \mathbb {R} } , and g i : R → R {\displaystyle g_{i}:\mathbb {R} \to \mathbb {R} } for each i = 1 , 2 , … , k . {\displaystyle i=1,2,\dots ,k.} As this case occurs often in

3315-642: Is sometimes abbreviated as ( f ∘ g ) ′ = ( f ′ ∘ g ) ⋅ g ′ . {\displaystyle (f\circ g)'=(f'\circ g)\cdot g'.} If y = f ( u ) and u = g ( x ) , then this abbreviated form is written in Leibniz notation as: d y d x = d y d u ⋅ d u d x . {\displaystyle {\frac {dy}{dx}}={\frac {dy}{du}}\cdot {\frac {du}{dx}}.} The points where

3400-575: Is the ratio of the speeds, and the speed is the derivative of the position with respect to the time; that is, d z d x = d z d t d x d t , {\displaystyle {\frac {dz}{dx}}={\frac {\frac {dz}{dt}}{\frac {dx}{dt}}},} or, equivalently, d z d t = d z d x ⋅ d x d t , {\displaystyle {\frac {dz}{dt}}={\frac {dz}{dx}}\cdot {\frac {dx}{dt}},} which

3485-414: Is the usual formula for the quotient rule. Suppose that y = g ( x ) has an inverse function . Call its inverse function f so that we have x = f ( y ) . There is a formula for the derivative of f in terms of the derivative of g . To see this, note that f and g satisfy the formula f ( g ( x ) ) = x . {\displaystyle f(g(x))=x.} And because

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3570-399: Is therefore: d y d x = e sin ⁡ ( x 2 ) ⋅ cos ⁡ ( x 2 ) ⋅ 2 x . {\displaystyle {\frac {dy}{dx}}=e^{\sin(x^{2})}\cdot \cos(x^{2})\cdot 2x.} Another way of computing this derivative is to view the composite function f ∘ g ∘ h as

3655-450: The barycentric radial-velocity measure or spectroscopic radial velocity. However, due to relativistic and cosmological effects over the great distances that light typically travels to reach the observer from an astronomical object, this measure cannot be accurately transformed to a geometric radial velocity without additional assumptions about the object and the space between it and the observer. By contrast, astrometric radial velocity

3740-526: The data reduction is to remove the contributions of Chain rule In calculus , the chain rule is a formula that expresses the derivative of the composition of two differentiable functions f and g in terms of the derivatives of f and g . More precisely, if h = f ∘ g {\displaystyle h=f\circ g} is the function such that h ( x ) = f ( g ( x ) ) {\displaystyle h(x)=f(g(x))} for every x , then

3825-484: The standard part we obtain d y d t = d y d x d x d t {\displaystyle {\frac {dy}{dt}}={\frac {dy}{dx}}{\frac {dx}{dt}}} which is the chain rule. The full generalization of the chain rule to multi-variable functions (such as f : R m → R n {\displaystyle f:\mathbb {R} ^{m}\to \mathbb {R} ^{n}} )

3910-1038: The Lagrange notation, f 1 . . n ′ ( x ) = f 1 ′ ( f 2 . . n ( x ) ) f 2 ′ ( f 3 . . n ( x ) ) ⋯ f n − 1 ′ ( f n . . n ( x ) ) f n ′ ( x ) = ∏ k = 1 n f k ′ ( f ( k + 1 . . n ) ( x ) ) {\displaystyle f_{1\,.\,.\,n}'(x)=f_{1}'\left(f_{2\,.\,.\,n}(x)\right)\;f_{2}'\left(f_{3\,.\,.\,n}(x)\right)\cdots f_{n-1}'\left(f_{n\,.\,.\,n}(x)\right)\;f_{n}'(x)=\prod _{k=1}^{n}f_{k}'\left(f_{(k+1\,.\,.\,n)}(x)\right)} The chain rule can be used to derive some well-known differentiation rules. For example,

3995-613: The Sun, based on observed redshift of the star's light. In many binary stars , the orbital motion usually causes radial velocity variations of several kilometres per second (km/s). As the spectra of these stars vary due to the Doppler effect, they are called spectroscopic binaries . Radial velocity can be used to estimate the ratio of the masses of the stars, and some orbital elements , such as eccentricity and semimajor axis . The same method has also been used to detect planets around stars, in

4080-399: The advantage that it generalizes to several variables. It relies on the following equivalent definition of differentiability at a point: A function g is differentiable at a if there exists a real number g ′( a ) and a function ε ( h ) that tends to zero as h tends to zero, and furthermore g ( a + h ) − g ( a ) = g ′ (

4165-1340: The chain rule again. For concreteness, consider the function y = e sin ⁡ ( x 2 ) . {\displaystyle y=e^{\sin(x^{2})}.} This can be decomposed as the composite of three functions: y = f ( u ) = e u , u = g ( v ) = sin ⁡ v , v = h ( x ) = x 2 . {\displaystyle {\begin{aligned}y&=f(u)=e^{u},\\u&=g(v)=\sin v,\\v&=h(x)=x^{2}.\end{aligned}}} So that y = f ( g ( h ( x ) ) ) {\displaystyle y=f(g(h(x)))} . Their derivatives are: d y d u = f ′ ( u ) = e u , d u d v = g ′ ( v ) = cos ⁡ v , d v d x = h ′ ( x ) = 2 x . {\displaystyle {\begin{aligned}{\frac {dy}{du}}&=f'(u)=e^{u},\\{\frac {du}{dv}}&=g'(v)=\cos v,\\{\frac {dv}{dx}}&=h'(x)=2x.\end{aligned}}} The chain rule states that

4250-401: The chain rule and the fact that differentiable functions and compositions of continuous functions are continuous, we have that there exist functions q , continuous at g ( a ) , and r , continuous at a , and such that, f ( g ( x ) ) − f ( g ( a ) ) = q ( g ( x ) ) ( g ( x ) − g (

4335-472: The chain rule appears in Lagrange's 1797 Théorie des fonctions analytiques ; it also appears in Cauchy's 1823 Résumé des Leçons données a L’École Royale Polytechnique sur Le Calcul Infinitesimal . The simplest form of the chain rule is for real-valued functions of one real variable. It states that if g is a function that is differentiable at a point c (i.e. the derivative g ′( c ) exists) and f

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4420-405: The chain rule begins by defining the derivative of the composite function f ∘ g , where we take the limit of the difference quotient for f ∘ g as x approaches a : ( f ∘ g ) ′ ( a ) = lim x → a f ( g ( x ) ) − f ( g ( a ) ) x −

4505-409: The chain rule for the composition of functions x ↦ f ( g 1 ( x ) , … , g k ( x ) ) , {\displaystyle x\mapsto f(g_{1}(x),\dots ,g_{k}(x)),} one needs the partial derivatives of f with respect to its k arguments. The usual notations for partial derivatives involve names for the arguments of

4590-694: The chain rule gives d d x ( g ( x ) + h ( x ) ) = ( d d x g ( x ) ) D 1 f + ( d d x h ( x ) ) D 2 f = d d x g ( x ) + d d x h ( x ) . {\displaystyle {\frac {d}{dx}}(g(x)+h(x))=\left({\frac {d}{dx}}g(x)\right)D_{1}f+\left({\frac {d}{dx}}h(x)\right)D_{2}f={\frac {d}{dx}}g(x)+{\frac {d}{dx}}h(x).} For multiplication f ( u , v ) = u v , {\displaystyle f(u,v)=uv,}

4675-580: The chain rule is d d x f ( g 1 ( x ) , … , g k ( x ) ) = ∑ i = 1 k ( d d x g i ( x ) ) D i f ( g 1 ( x ) , … , g k ( x ) ) . {\displaystyle {\frac {d}{dx}}f(g_{1}(x),\dots ,g_{k}(x))=\sum _{i=1}^{k}\left({\frac {d}{dx}}{g_{i}}(x)\right)D_{i}f(g_{1}(x),\dots ,g_{k}(x)).} If

4760-613: The chain rule is, in Lagrange's notation , h ′ ( x ) = f ′ ( g ( x ) ) g ′ ( x ) . {\displaystyle h'(x)=f'(g(x))g'(x).} or, equivalently, h ′ = ( f ∘ g ) ′ = ( f ′ ∘ g ) ⋅ g ′ . {\displaystyle h'=(f\circ g)'=(f'\circ g)\cdot g'.} The chain rule may also be expressed in Leibniz's notation . If

4845-431: The composite of f ∘ g and h . Applying the chain rule in this manner would yield: ( f ∘ g ∘ h ) ′ ( a ) = ( f ∘ g ) ′ ( h ( a ) ) ⋅ h ′ ( a ) = f ′ ( g ( h ( a ) ) ) ⋅ g ′ ( h (

4930-683: The corresponding Δ x = g ( t + Δ t ) − g ( t ) {\displaystyle \Delta x=g(t+\Delta t)-g(t)} and then the corresponding Δ y = f ( x + Δ x ) − f ( x ) {\displaystyle \Delta y=f(x+\Delta x)-f(x)} , so that Δ y Δ t = Δ y Δ x Δ x Δ t {\displaystyle {\frac {\Delta y}{\Delta t}}={\frac {\Delta y}{\Delta x}}{\frac {\Delta x}{\Delta t}}} and applying

5015-958: The derivative of 1/ g ( x ) , notice that it is the composite of g with the reciprocal function, that is, the function that sends x to 1/ x . The derivative of the reciprocal function is − 1 / x 2 {\displaystyle -1/x^{2}\!} . By applying the chain rule, the last expression becomes: f ′ ( x ) ⋅ 1 g ( x ) + f ( x ) ⋅ ( − 1 g ( x ) 2 ⋅ g ′ ( x ) ) = f ′ ( x ) g ( x ) − f ( x ) g ′ ( x ) g ( x ) 2 , {\displaystyle f'(x)\cdot {\frac {1}{g(x)}}+f(x)\cdot \left(-{\frac {1}{g(x)^{2}}}\cdot g'(x)\right)={\frac {f'(x)g(x)-f(x)g'(x)}{g(x)^{2}}},} which

5100-440: The derivative of a composite of more than two functions, notice that the composite of f , g , and h (in that order) is the composite of f with g ∘ h . The chain rule states that to compute the derivative of f ∘ g ∘ h , it is sufficient to compute the derivative of f and the derivative of g ∘ h . The derivative of f can be calculated directly, and the derivative of g ∘ h can be calculated by applying

5185-411: The derivative of their composite at the point x = a is: ( f ∘ g ∘ h ) ′ ( a ) = f ′ ( ( g ∘ h ) ( a ) ) ⋅ ( g ∘ h ) ′ ( a ) = f ′ ( ( g ∘ h ) (

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5270-480: The derivatives are evaluated may also be stated explicitly: d y d x | x = c = d y d u | u = g ( c ) ⋅ d u d x | x = c . {\displaystyle \left.{\frac {dy}{dx}}\right|_{x=c}=\left.{\frac {dy}{du}}\right|_{u=g(c)}\cdot \left.{\frac {du}{dx}}\right|_{x=c}.} Carrying

5355-429: The derivatives have to be evaluated. In integration , the counterpart to the chain rule is the substitution rule . Intuitively, the chain rule states that knowing the instantaneous rate of change of z relative to y and that of y relative to x allows one to calculate the instantaneous rate of change of z relative to x as the product of the two rates of change. As put by George F. Simmons : "If

5440-427: The difference quotient for f ∘ g is always equal to: Q ( g ( x ) ) ⋅ g ( x ) − g ( a ) x − a . {\displaystyle Q(g(x))\cdot {\frac {g(x)-g(a)}{x-a}}.} Whenever g ( x ) is not equal to g ( a ) , this is clear because the factors of g ( x ) − g ( a ) cancel. When g ( x ) equals g (

5525-411: The differentiability of a function can be used to give an elegant proof of the chain rule. Under this definition, a function f is differentiable at a point a if and only if there is a function q , continuous at a and such that f ( x ) − f ( a ) = q ( x )( x − a ) . There is at most one such function, and if f is differentiable at a then f ′( a ) = q ( a ) . Given the assumptions of

5610-2575: The first few derivatives are: d y d x = d y d u d u d x d 2 y d x 2 = d 2 y d u 2 ( d u d x ) 2 + d y d u d 2 u d x 2 d 3 y d x 3 = d 3 y d u 3 ( d u d x ) 3 + 3 d 2 y d u 2 d u d x d 2 u d x 2 + d y d u d 3 u d x 3 d 4 y d x 4 = d 4 y d u 4 ( d u d x ) 4 + 6 d 3 y d u 3 ( d u d x ) 2 d 2 u d x 2 + d 2 y d u 2 ( 4 d u d x d 3 u d x 3 + 3 ( d 2 u d x 2 ) 2 ) + d y d u d 4 u d x 4 . {\displaystyle {\begin{aligned}{\frac {dy}{dx}}&={\frac {dy}{du}}{\frac {du}{dx}}\\{\frac {d^{2}y}{dx^{2}}}&={\frac {d^{2}y}{du^{2}}}\left({\frac {du}{dx}}\right)^{2}+{\frac {dy}{du}}{\frac {d^{2}u}{dx^{2}}}\\{\frac {d^{3}y}{dx^{3}}}&={\frac {d^{3}y}{du^{3}}}\left({\frac {du}{dx}}\right)^{3}+3\,{\frac {d^{2}y}{du^{2}}}{\frac {du}{dx}}{\frac {d^{2}u}{dx^{2}}}+{\frac {dy}{du}}{\frac {d^{3}u}{dx^{3}}}\\{\frac {d^{4}y}{dx^{4}}}&={\frac {d^{4}y}{du^{4}}}\left({\frac {du}{dx}}\right)^{4}+6\,{\frac {d^{3}y}{du^{3}}}\left({\frac {du}{dx}}\right)^{2}{\frac {d^{2}u}{dx^{2}}}+{\frac {d^{2}y}{du^{2}}}\left(4\,{\frac {du}{dx}}{\frac {d^{3}u}{dx^{3}}}+3\,\left({\frac {d^{2}u}{dx^{2}}}\right)^{2}\right)+{\frac {dy}{du}}{\frac {d^{4}u}{dx^{4}}}.\end{aligned}}} One proof of

5695-489: The first proof is played by η in this proof. They are related by the equation: Q ( y ) = f ′ ( g ( a ) ) + η ( y − g ( a ) ) . {\displaystyle Q(y)=f'(g(a))+\eta (y-g(a)).} The need to define Q at g ( a ) is analogous to the need to define η at zero. Constantin Carathéodory 's alternative definition of

5780-416: The first two bracketed terms tend to zero as h tends to zero. Applying the same theorem on products of limits as in the first proof, the third bracketed term also tends zero. Because the above expression is equal to the difference f ( g ( a + h )) − f ( g ( a )) , by the definition of the derivative f ∘ g is differentiable at a and its derivative is f ′( g ( a )) g ′( a ). The role of Q in

5865-408: The formula fails in this case. This is not surprising because f is not differentiable at zero. The chain rule forms the basis of the back propagation algorithm, which is used in gradient descent of neural networks in deep learning ( artificial intelligence ). Faà di Bruno's formula generalizes the chain rule to higher derivatives. Assuming that y = f ( u ) and u = g ( x ) , then

5950-431: The function g ( x ) = e . It has an inverse f ( y ) = ln y . Because g ′( x ) = e , the above formula says that d d y ln ⁡ y = 1 e ln ⁡ y = 1 y . {\displaystyle {\frac {d}{dy}}\ln y={\frac {1}{e^{\ln y}}}={\frac {1}{y}}.} This formula is true whenever g is differentiable and its inverse f

6035-499: The function f is addition, that is, if f ( u , v ) = u + v , {\displaystyle f(u,v)=u+v,} then D 1 f = ∂ f ∂ u = 1 {\textstyle D_{1}f={\frac {\partial f}{\partial u}}=1} and D 2 f = ∂ f ∂ v = 1 {\textstyle D_{2}f={\frac {\partial f}{\partial v}}=1} . Thus,

6120-423: The function. As these arguments are not named in the above formula, it is simpler and clearer to use D -Notation , and to denote by D i f {\displaystyle D_{i}f} the partial derivative of f with respect to its i th argument, and by D i f ( z ) {\displaystyle D_{i}f(z)} the value of this derivative at z . With this notation,

6205-544: The functions f ( g ( x ) ) {\displaystyle f(g(x))} and x are equal, their derivatives must be equal. The derivative of x is the constant function with value 1, and the derivative of f ( g ( x ) ) {\displaystyle f(g(x))} is determined by the chain rule. Therefore, we have that: f ′ ( g ( x ) ) g ′ ( x ) = 1. {\displaystyle f'(g(x))g'(x)=1.} To express f' as

6290-418: The inner product is either +1 or -1, for parallel and antiparallel vectors , respectively. A singularity exists for coincident observer target, i.e., r = 0 {\displaystyle r=0} ; in this case, range rate is undefined. In astronomy, radial velocity is often measured to the first order of approximation by Doppler spectroscopy . The quantity obtained by this method may be called

6375-398: The limit of a product exists if the limits of its factors exist. When this happens, the limit of the product of these two factors will equal the product of the limits of the factors. The two factors are Q ( g ( x )) and ( g ( x ) − g ( a )) / ( x − a ) . The latter is the difference quotient for g at a , and because g is differentiable at a by assumption, its limit as x tends to

6460-949: The quotient rule is a consequence of the chain rule and the product rule . To see this, write the function f ( x )/ g ( x ) as the product f ( x ) · 1/ g ( x ) . First apply the product rule: d d x ( f ( x ) g ( x ) ) = d d x ( f ( x ) ⋅ 1 g ( x ) ) = f ′ ( x ) ⋅ 1 g ( x ) + f ( x ) ⋅ d d x ( 1 g ( x ) ) . {\displaystyle {\begin{aligned}{\frac {d}{dx}}\left({\frac {f(x)}{g(x)}}\right)&={\frac {d}{dx}}\left(f(x)\cdot {\frac {1}{g(x)}}\right)\\&=f'(x)\cdot {\frac {1}{g(x)}}+f(x)\cdot {\frac {d}{dx}}\left({\frac {1}{g(x)}}\right).\end{aligned}}} To compute

6545-425: The radial velocity then denotes the speed with which the object moves away from the Earth (or approaches it, for a negative radial velocity). Given a differentiable vector r ∈ R 3 {\displaystyle \mathbf {r} \in \mathbb {R} ^{3}} defining the instantaneous relative position of a target with respect to an observer. Let the instantaneous relative velocity of

6630-482: The range rate is simply expressed as i.e., the projection of the relative velocity vector onto the LOS direction. Further defining the velocity direction v ^ = v / v {\displaystyle {\hat {v}}=\mathbf {v} /{v}} , with the relative speed v = ‖ v ‖ {\displaystyle v=\|\mathbf {v} \|} , we have: where

6715-518: The right-hand-side by the chain rule using ( 1 ) the expression becomes By reciprocity, ⟨ v , r ⟩ = ⟨ r , v ⟩ {\displaystyle \langle \mathbf {v} ,\mathbf {r} \rangle =\langle \mathbf {r} ,\mathbf {v} \rangle } . Defining the unit relative position vector r ^ = r / r {\displaystyle {\hat {r}}=\mathbf {r} /{r}} (or LOS direction),

6800-533: The same reasoning further, given n functions f 1 , … , f n {\displaystyle f_{1},\ldots ,f_{n}\!} with the composite function f 1 ∘ ( f 2 ∘ ⋯ ( f n − 1 ∘ f n ) ) {\displaystyle f_{1}\circ (f_{2}\circ \cdots (f_{n-1}\circ f_{n}))\!} , if each function f i {\displaystyle f_{i}\!}

6885-420: The star. When the star moves towards us, its spectrum is blueshifted, while it is redshifted when it moves away from us. By regularly looking at the spectrum of a star—and so, measuring its velocity—it can be determined if it moves periodically due to the influence of an exoplanet companion. From the instrumental perspective, velocities are measured relative to the telescope's motion. So an important first step of

6970-480: The study of functions of a single variable, it is worth describing it separately. Let f : R k → R {\displaystyle f:\mathbb {R} ^{k}\to \mathbb {R} } , and g i : R → R {\displaystyle g_{i}:\mathbb {R} \to \mathbb {R} } for each i = 1 , 2 , … , k . {\displaystyle i=1,2,\dots ,k.} To write

7055-410: The target with respect to the observer be The magnitude of the position vector r {\displaystyle \mathbf {r} } is defined as in terms of the inner product The quantity range rate is the time derivative of the magnitude ( norm ) of r {\displaystyle \mathbf {r} } , expressed as Substituting ( 2 ) into ( 3 ) Evaluating the derivative of

7140-423: The two points. The radial speed or range rate is the temporal rate of the distance or range between the two points. It is a signed scalar quantity , formulated as the scalar projection of the relative velocity vector onto the LOS direction. Equivalently, radial speed equals the norm of the radial velocity, modulo the sign. In astronomy, the point is usually taken to be the observer on Earth, so

7225-399: The way that the movement's measurement determines the planet's orbital period, while the resulting radial-velocity amplitude allows the calculation of the lower bound on a planet's mass using the binary mass function . Radial velocity methods alone may only reveal a lower bound, since a large planet orbiting at a very high angle to the line of sight will perturb its star radially as much as

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