Question

What is the chain rule?

Answer 1

The chain rule for differentiation is essentially:

`(dy)/(dx) = (dy)/(du) * (du)/(dx)`

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

Explanation:

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)`