# Given that tany= x^2, what is the value of dy/dx?

Dec 7, 2016

$\frac{\mathrm{dy}}{\mathrm{dx}} = \frac{2 x}{\sec} ^ 2 y$

#### Explanation:

We write $\tan y$ as $\sin \frac{y}{\cos} y$ and differentiate using the quotient rule (with respect to $x$) .

$\sin \frac{y}{\cos} y = {x}^{2}$

$\frac{\cos y \left(\cos y\right) \left(\frac{\mathrm{dy}}{\mathrm{dx}}\right) - \left(- \sin y \times \sin y\right) \frac{\mathrm{dy}}{\mathrm{dx}}}{\cos y} ^ 2 = 2 x$

$\frac{{\cos}^{2} y \left(\frac{\mathrm{dy}}{\mathrm{dx}}\right) + {\sin}^{2} y \left(\frac{\mathrm{dy}}{\mathrm{dx}}\right)}{\cos} ^ 2 y = 2 x$

We use the identity ${\sin}^{2} \beta + {\cos}^{2} \beta = 1$ to simplify further...

$\frac{\frac{\mathrm{dy}}{\mathrm{dx}}}{{\cos}^{2} y} = 2 x$

Use the identity $\sec \theta = \frac{1}{\cos} \theta$...

$\frac{\mathrm{dy}}{\mathrm{dx}} {\sec}^{2} y = 2 x$

$\frac{\mathrm{dy}}{\mathrm{dx}} = \frac{2 x}{\sec} ^ 2 y$

Hopefully this helps!