When do we use the chain rule and why?

Question

I need help!!

Answer 1

The chain rule is used primarily to break down complex or multivariable functions.

For example, take:
f(x) = (5x^3 + 3x^2 - 9x + 12)^9

First redefine the function as f(x) = h(g(x)) where g(x) = 5x^3 + 3x^2 - 9x + 12, and h(x) = x^9.
f'x = h'(g(x)) * g'(x) (chain rule)
f'x = 9(g(x))^8 * (15x^2 + 6x - 9)
f'x = 9(5x^3 + 3x^2 - 9x + 12)^8 * (15x^2 + 6x - 9)