How do you differentiate f(x) = sqrt((5x+1)^2+(2x-1)) using the chain rule?

Nov 11, 2015

$f ' \left(x\right) = \frac{25 x + 6}{\sqrt{25 {x}^{2} + 12 x}}$

Explanation:

After simplifying,

$f \left(x\right) = \sqrt{25 {x}^{2} + 12 x}$.

Let $u = 25 {x}^{2} + 12 x$.

Then, $\frac{\mathrm{du}}{\mathrm{dx}} = 50 x + 12$.

Using the chain rule,

$f ' \left(x\right) = \frac{d}{\mathrm{dx}} \left(\sqrt{25 {x}^{2} + 12 x}\right)$

$= \frac{d}{\mathrm{dx}} \left(\sqrt{u}\right)$

$= \frac{d}{\mathrm{du}} \left(\sqrt{u}\right) \frac{\mathrm{du}}{\mathrm{dx}}$

$= \frac{1}{2 \sqrt{u}} \left(50 x + 12\right)$

$= \frac{25 x + 6}{\sqrt{25 {x}^{2} + 12 x}}$.