# How do you differentiate f(x) = x^3sqrt(7x+1) using the product rule?

Apr 13, 2017

$f ' \left(x\right) = 3 {x}^{2} \sqrt{7 x + 1} + \frac{7 {x}^{3}}{2 \sqrt{7 x + 1}}$

#### Explanation:

By rewriting a bit,

$f \left(x\right) = {x}^{3} {\left(7 x + 1\right)}^{\frac{1}{2}}$

By Product Rule and Chain Rule,

$f ' \left(x\right) = 3 {x}^{2} \cdot {\left(7 x + 1\right)}^{\frac{1}{2}} + {x}^{3} \cdot \frac{1}{2} {\left(7 x + 1\right)}^{- \frac{1}{2}} \left(7\right)$

By cleaning up a bit,

$= 3 {x}^{2} \sqrt{7 x + 1} + \frac{7 {x}^{3}}{2 \sqrt{7 x + 1}}$

I hope that this was clear.