What is the chain rule?

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Answer 1 · 2015-11-03T22:23:38

The chain rule for differentiation is essentially:

$\frac{d y}{d x} = \frac{d y}{d u} \cdot \frac{d u}{d x}$ $(dy)/(dx) = (dy)/(du) * (du)/(dx)$

So if $y = f (g (x))$ $y = f(g(x))$ then $d/(dx) y = f'(g(x)) * g'(x)$

For example, suppose $y = (x^2+x-1)^10$

Let $u = x^2+x-1$

Then $y = u^10$ and using the power rule:

$(dy)/(du) = 10 u^9$

$(du)/(dx) = 2x+1$

Hence:

$(dy)/(dx) = (dy)/(du) * (du)/(dx) = 10u^9 * (2x+1) = 10(x^2+x-1)^9(2x+1)$

Or we could formulate this as follows:

$g(x) = x^2+x-1$

$f(u) = u^10$

$y = f(g(x)) = (x^2+x-1)^10$

$d/(dx) y = f'(g(x)) * g'(x) = 10(g(x))^9 * (2x+1)$

$=10(x^2+x+1)^9(2x+1)$