Question

How is the chain rule different from the product rule?

Answer 1

The chain rule is used to differentiate compositions of functions. The product rule is used to differentiate products of function.

Explanation:

Composition and product are different operations.

For `f(x) = 2x+3` and `g(x) = 5x+7`, the composition

`(f@g)(x) = f(g(x)) = f(5x+7) = 2(5x+7)+3 = 10x + 17`

is not at all the same as the product

`(fg)(x) = f(x)g(x) = (2x+3)(5x+7) = 10x^2 +29x+21`.