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In calculus, the chain rule is a formula for the derivative of the composite of two functions. ## Informal discussion

## Theorem

The chain rule in one variable may be stated more completely as follows. Let g be a real-valued function on (a,b) which is differentiable at c ∈ (a,b); and f a real-valued function defined on an interval I containing the range of g and g(c) as an interior point. If f is differentiable at g(c), then## Examples

### Example I

Suppose that a mountain climber ascends at a rate of 0.5 kilometers per hour. The temperature is lower at higher elevations; suppose the rate by which it decreases is 6 °C per kilometer. If one multiplies 6 °C per kilometer by 0.5 kilometer per hour, one obtains 3 °C per hour. This calculation is a typical chain rule application.
### Example II

Consider the function f(x) = (x^{2} + 1)^{3}. Since f(x) = h(g(x)) where g(x) = x^{2} + 1 and h(x) = x^{3} it follows from the chain rule that

In intuitive terms, if a variable, y, depends on a second variable, u, which in turn depends on a third variable, x, then the rate of change of y with respect to x can be computed as the rate of change of y with respect to u multiplied by the rate of change of u with respect to x.

- For an explanation of notation used in this section, see Function composition.

- $(f\; circ\; g)\text{'}(x)\; =\; f\text{'}(g(x))\; g\text{'}(x),,$

which in short form is written as

- $(f\; circ\; g)\text{'}\; =\; f\text{'}circ\; gcdot\; g\text{'}.$

Alternatively, in the Leibniz notation, the chain rule is

- $frac\; \{dy\}\{dx\}\; =\; frac\; \{dy\}\; \{du\}\; cdotfrac\; \{du\}\{dx\}.$

In integration, the counterpart to the chain rule is the substitution rule.

- $(fcirc\; g)(x)$ is differentiable at x = c, and
- $(fcirc\; g)\text{'}(c)\; =\; f\text{'}(g(c))g\text{'}(c).$

$f\; \text{'}(x)\; ,$ $=\; h\; \text{'}(g(x))\; g\; \text{'}\; (x)\; ,$ $=\; 3(g(x))^2(2x)\; ,$ $=\; 3(x^2\; +\; 1)^2(2x)\; ,$ $=\; 6x(x^2\; +\; 1)^2.\; ,$ In order to differentiate the trigonometric function

- $f(x)\; =\; sin(x^2),,$

^{2}. The chain rule then yields- $f\text{'}(x)\; =\; 2x\; cos(x^2)\; ,$

^{2}) and g′(x) = 2x.### Example III

Differentiate arctan(sin x).- $frac\{d\}\{dx\}arctan\; x\; =\; frac\{1\}\{1+x^2\}$

Thus, by the chain rule,

- $frac\{d\}\{dx\}arctan\; f(x)\; =\; frac\{f\text{'}(x)\}\{1+f^2(x)\},,$

and in particular,

- $frac\{d\}\{dx\}arctan(sin\; x)\; =\; frac\{cos\; x\}\{1+sin^2\; x\},.$

## Chain rule for several variables

The chain rule works for functions of more than one variable. Consider the function z = f(x, y) where x = g(t) and y = h(t), and g(t) and h(t) are differentiable with respect to t, then- $\{\; dz\; over\; dt\}=\{partial\; z\; over\; partial\; x\}\{dx\; over\; dt\}+\{partial\; z\; over\; partial\; y\}\{dy\; over\; dt\}.$

Suppose that each argument of z = f(u, v) is a two-variable function such that u = h(x, y) and v = g(x, y), and that these functions are all differentiable. Then the chain rule would look like:

- $\{partial\; z\; over\; partial\; x\}=\{partial\; z\; over\; partial\; u\}\{partial\; u\; over\; partial\; x\}+\{partial\; z\; over\; partial\; v\}\{partial\; v\; over\; partial\; x\}$

- $\{partial\; z\; over\; partial\; y\}=\{partial\; z\; over\; partial\; u\}\{partial\; u\; over\; partial\; y\}+\{partial\; z\; over\; partial\; v\}\{partial\; v\; over\; partial\; y\}.$

If we considered

- $vec\; r\; =\; (u,v)$

- $frac\{partial\; f\}\{partial\; x\}=vec\; nabla\; f\; cdot\; frac\{partial\; vec\; r\}\{partial\; x\}.$

More generally, for functions of vectors to vectors, the chain rule says that the Jacobian matrix of a composite function is the product of the Jacobian matrices of the two functions:

- $frac\{partial(z\_1,ldots,z\_m)\}\{partial(x\_1,ldots,x\_p)\}\; =\; frac\{partial(z\_1,ldots,z\_m)\}\{partial(y\_1,ldots,y\_n)\}\; frac\{partial(y\_1,ldots,y\_n)\}\{partial(x\_1,ldots,x\_p)\}.$

## Proof of the chain rule

Let f and g be functions and let x be a number such that f is differentiable at g(x) and g is differentiable at x. Then by the definition of differentiability,- $g(x+delta)-g(x)=\; delta\; g\text{'}(x)\; +\; epsilon(delta)delta\; ,$

- $f(g(x)+alpha)\; -\; f(g(x))\; =\; alpha\; f\text{'}(g(x))\; +\; eta(alpha)alpha\; ,$

Now

$f(g(x+delta))-f(g(x)),$ $=\; f(g(x)\; +\; delta\; g\text{'}(x)+epsilon(delta)delta)\; -\; f(g(x))\; ,$ $=\; alpha\_delta\; f\text{'}(g(x))\; +\; eta(alpha\_delta)alpha\_delta\; ,$ where

- $alpha\_delta\; =\; delta\; g\text{'}(x)\; +\; epsilon(delta)delta.\; ,$

_{δ}/δ → g′(x) and α_{δ}→ 0, and thus η(α_{δ}) → 0. It follows that- $frac\{f(g(x+delta))-f(g(x))\}\{delta\}\; to\; g\text{'}(x)f\text{'}(g(x))mbox\{\; as\; \}\; delta\; to\; 0.$

## The fundamental chain rule

The chain rule is a fundamental property of all definitions of derivative and is therefore valid in much more general contexts. For instance, if E, F and G are Banach spaces (which includes Euclidean space) and f : E → F and g : F → G are functions, and if x is an element of E such that f is differentiable at x and g is differentiable at f(x), then the derivative (the Fréchet derivative) of the composition g o f at the point x is given by- $mbox\{D\}\_xleft(g\; circ\; fright)\; =\; mbox\{D\}\_\{fleft(xright)\}left(gright)\; circ\; mbox\{D\}\_xleft(fright).$

Note that the derivatives here are linear maps and not numbers. If the linear maps are represented as matrices (namely Jacobians), the composition on the right hand side turns into a matrix multiplication.

A particularly clear formulation of the chain rule can be achieved in the most general setting: let M, N and P be C

^{k}manifolds (or even Banach-manifolds) and let- f : M → N and g : N → P

be differentiable maps. The derivative of f, denoted by df, is then a map from the tangent bundle of M to the tangent bundle of N, and we may write

- $mbox\{d\}left(g\; circ\; fright)\; =\; mbox\{d\}g\; circ\; mbox\{d\}f.$

In this way, the formation of derivatives and tangent bundles is seen as a functor on the category of C

^{∞}manifolds with C^{∞}maps as morphisms.## Tensors and the chain rule

See tensor field for an advanced explanation of the fundamental role the chain rule plays in the geometric nature of tensors.## Higher derivatives

Faà di Bruno's formula generalizes the chain rule to higher derivatives. The first few derivatives are- $frac\{d\; (f\; circ\; g)\; \}\{dx\}\; =\; frac\{df\}\{dg\}frac\{dg\}\{dx\}$

- $$

- $$

- $$

+ frac{df}{dg}frac{d^4 g}{dx^4}.## See also

- Inverse chain rule
- Triple product rule
- Derivative
- Leibniz integral rule
- Leibniz rule (generalized product rule)

## References

## External links

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Last updated on Thursday October 02, 2008 at 11:27:56 PDT (GMT -0700)

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