Difference between revisions of "Chain rule for higher derivatives"

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(Created page with "==Statement== Suppose <math>n</math> is a natural number, and <math>f</math> and <math>g</math> are functions such that <math>g</math> is <math>n</math> times differentiable at ...")
 
(Particular cases)
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! Value of <math>n</math> !! Formula for <math>n^{th}</math> derivative of <math>f \circ g</math> at <math>x_0</math>
 
! Value of <math>n</math> !! Formula for <math>n^{th}</math> derivative of <math>f \circ g</math> at <math>x_0</math>
 
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| 1 || <math>f'(g(x_0))g'(x_0)</math> (this is the [[chain rule for differentiation]])
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| 1 || <math>\! f'(g(x_0))g'(x_0)</math> (this is the [[chain rule for differentiation]])
 
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| 2 || <math>f''(g(x_0))(g'(x_0))^2 + f'(g(x_0))g''(x_0)</math> (obtained by using the [[chain rule for differentiation]] twice ''and'' using the [[product rule for differentiation]]).
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| 2 || <math>\! f''(g(x_0))(g'(x_0))^2 + f'(g(x_0))g''(x_0)</math> (obtained by using the [[chain rule for differentiation]] twice ''and'' using the [[product rule for differentiation]]).
 
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Revision as of 13:42, 26 August 2011

Statement

Suppose n is a natural number, and f and g are functions such that g is n times differentiable at x = x_0 and f is n times differentiable at g(x_0). Then, f \circ g is n times differentiable at x_0. Further, the value of the n^{th} derivative is given by a complicated formula involving compositions, products, derivatives, evaluations, and sums that depends on n.

Particular cases

Value of n Formula for n^{th} derivative of f \circ g at x_0
1 \! f'(g(x_0))g'(x_0) (this is the chain rule for differentiation)
2 \! f''(g(x_0))(g'(x_0))^2 + f'(g(x_0))g''(x_0) (obtained by using the chain rule for differentiation twice and using the product rule for differentiation).