# How is the chain rule different from the product rule?

Mar 14, 2016

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 $f \left(x\right) = 2 x + 3$ and $g \left(x\right) = 5 x + 7$, the composition

$\left(f \circ g\right) \left(x\right) = f \left(g \left(x\right)\right) = f \left(5 x + 7\right) = 2 \left(5 x + 7\right) + 3 = 10 x + 17$

is not at all the same as the product

$\left(f g\right) \left(x\right) = f \left(x\right) g \left(x\right) = \left(2 x + 3\right) \left(5 x + 7\right) = 10 {x}^{2} + 29 x + 21$.