# How do you differentiate g(y) =(4x^2+5)(x^2 - 1)  using the product rule?

Jun 19, 2016

The product rule states that for a function $f \left(x\right) = g \left(x\right) h \left(x\right)$, $f ' \left(x\right)$ is given by $g ' \left(x\right) \times h \left(x\right) + g \left(x\right) \times h ' \left(x\right)$.

#### Explanation:

Let $4 {x}^{2} + 5$ be $g \left(x\right)$ and ${x}^{2} - 1$ be $h \left(x\right)$ and your whole function be $f \left(x\right)$ (instead of g(y)).

$g ' \left(x\right) = 8 x$

$h ' \left(x\right) = 2 x$

Thus, we have that $f ' \left(x\right) = \left(8 x \left({x}^{2} - 1\right)\right) + \left(2 x \left(4 {x}^{2} + 5\right)\right)$

$f ' \left(x\right) = 8 {x}^{3} - 8 x + 8 {x}^{3} + 10 x$

$f ' \left(x\right) = 16 {x}^{3} + 2 x$

Hopefully this helps!