Difference between revisions of "Chain rule for higher derivatives"
From Calculus
(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]]) | + | | 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]]). | + | | 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 is a natural number, and and are functions such that is times differentiable at and is times differentiable at . Then, is times differentiable at . Further, the value of the derivative is given by a complicated formula involving compositions, products, derivatives, evaluations, and sums that depends on .
Particular cases
Value of | Formula for derivative of at |
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1 | (this is the chain rule for differentiation) |
2 | (obtained by using the chain rule for differentiation twice and using the product rule for differentiation). |