# How do you differentiate g(x) =x (4x^6 + 5)  using the product rule?

Apr 24, 2016

$28 {x}^{6} + 5$

#### Explanation:

The product rule says that if

$g \left(x\right) = a \left(x\right) \cdot b \left(x\right)$

then

$g ' \left(x\right) = a ' \left(x\right) b \left(x\right) + a \left(x\right) b ' \left(x\right)$

Substituting in from the question, $a \left(x\right) = x$ and $b \left(x\right) = 4 {x}^{6} + 5$, then

$a ' \left(x\right) = 1$

$b ' \left(x\right) = 24 {x}^{5}$

Now we can find that

$g ' \left(x\right) = 1 \left(4 {x}^{6} + 5\right) + x \left(24 {x}^{6}\right)$

and, on expanding the brackets,

$= 28 {x}^{6} + 5$

You can check this answer by distributing the function and just using the power rule:

$g \left(x\right) = 4 {x}^{7} + 5 x$

$g ' \left(x\right) = 28 {x}^{6} + 5$