# How do you differentiate y = (x + 7)^10 (x^2 + 2)^7?

Jan 18, 2016

See the explanation section, below.

#### Explanation:

For $y = f \cdot g$, we have $y ' = f ' g + f g '$

In this problem, $f = {\left(x + 7\right)}^{10}$, so $f ' = 10 {\left(x + 7\right)}^{9}$,

and $g = {\left({x}^{2} + 2\right)}^{7}$, so $g ' = 7 {\left({x}^{2} + 2\right)}^{6} \left(2 x\right)$ $\text{ }$ $\text{ }$ (chain rule).

Therefore,

$y ' = \left[10 {\left(x + 7\right)}^{9}\right] {\left({x}^{2} + 2\right)}^{7} + {\left(x + 7\right)}^{10} \left[7 {\left({x}^{2} + 2\right)}^{6} \left(2 x\right)\right]$.

Now use algebra to simplify as desired.

$y ' = 10 {\left(x + 7\right)}^{9} {\left({x}^{2} + 2\right)}^{7} + 14 x {\left(x + 7\right)}^{10} {\left({x}^{2} + 2\right)}^{6}$

$= 2 {\left(x + 7\right)}^{9} {\left({x}^{2} + 2\right)}^{6} \left(12 {x}^{2} + 49 x + 10\right)$.