# How do you differentiate f(x) = sqrt[ (3 x + 1) / (5 x^2 + 1) using the chain rule?

Jan 7, 2016

#### Answer:

We'll need chain rule here as well, besides the quotient rule itself.

#### Explanation:

• Chain rule: $\frac{\mathrm{dy}}{\mathrm{dx}} = \frac{\mathrm{dy}}{\mathrm{du}} \frac{\mathrm{du}}{\mathrm{dx}}$

• Quotient rule: $\left(\frac{a}{b}\right) ' = \frac{a ' b - a b '}{b} ^ 2$

Renaming $u = \frac{3 x + 1}{5 {x}^{2} + 1}$, we have $f \left(x\right) = \sqrt{u}$. Let's do it:

$\frac{\mathrm{dy}}{\mathrm{dx}} = \frac{1}{2 {u}^{\frac{1}{2}}} \frac{3 \left(5 {x}^{2} + 1\right) - \left(10 x \left(3 x + 1\right)\right)}{5 {x}^{2} + 1} ^ 2$

$\frac{\mathrm{dy}}{\mathrm{dx}} = \frac{1}{2 {u}^{\frac{1}{2}}} \frac{15 {x}^{2} + 3 - 30 {x}^{2} - 10 x}{5 {x}^{2} + 1} ^ 2$

(dy)/(dx)=(-15x^2-10x+3)/(2(5x^2+1)^2(sqrt((3x+1)/(5x^2+1)))