# How do you find the differential dy of the function y=sec^2x/(x^2+1)?

Oct 2, 2017

Compute the first derivative $\frac{\mathrm{dy}}{\mathrm{dx}}$, using the Quotient Rule .

Then multiply both sides by $\mathrm{dx}$

#### Explanation:

Given $y = {\sec}^{2} \frac{x}{{x}^{2} + 1}$

Applying the Quotient Rule :

$\frac{\mathrm{dy}}{\mathrm{dx}} = \frac{\left(\frac{d \left({\sec}^{2} \left(x\right)\right)}{\mathrm{dx}}\right) \left({x}^{2} + 1\right) - {\sec}^{2} \left(x\right) \left(\frac{d \left({x}^{2} + 1\right)}{\mathrm{dx}}\right)}{{x}^{2} + 1} ^ 2$

$\frac{\mathrm{dy}}{\mathrm{dx}} = \frac{2 \tan \left(x\right) {\sec}^{2} \left(x\right) \left({x}^{2} + 1\right) - 2 x {\sec}^{2} \left(x\right)}{{x}^{2} + 1} ^ 2$

Multiply both sides by $\mathrm{dx}$:

$\mathrm{dy} = \frac{2 \tan \left(x\right) {\sec}^{2} \left(x\right) \left({x}^{2} + 1\right) - 2 x {\sec}^{2} \left(x\right)}{{x}^{2} + 1} ^ 2 \mathrm{dx}$