# How do you find the derivative of y=10tan(20x)?

Jul 2, 2016

$\frac{\mathrm{dy}}{\mathrm{dx}} = 200 {\sec}^{2} \left(20\right) x$

#### Explanation:

First we need to find the derivative of $T a n a x$ where a is constant
$y = \tan a x$
$\frac{\mathrm{dy}}{\mathrm{dx}} = \frac{d}{\mathrm{dx}} \left(\frac{\sin a x}{\cos a x}\right)$
applying $\frac{u}{v}$ rule
where $u = \sin a x , v = \cos a x$
$\frac{\mathrm{dy}}{\mathrm{dx}}$=$\left(\frac{1}{{\cos}^{2} a x}\right)$(cosax$\left(\frac{d}{\mathrm{dx}} \left(\sin a x\right)\right)$-$\sin a x \left(\frac{d}{\mathrm{dx}} \left(\cos a x\right)\right)$
$\frac{\mathrm{dy}}{\mathrm{dx}}$=$\left(\frac{1}{{\cos}^{2} a x}\right)$*$\left(\cos a x \cdot a \cos x\right)$-sinax*(-asinax))
$\frac{\mathrm{dy}}{\mathrm{dx}}$=$a \frac{{\sin}^{2} a x + {\cos}^{2} a x}{{\cos}^{2} a x}$=$\frac{a}{{\cos}^{2} a x}$
$\frac{\mathrm{dy}}{\mathrm{dx}}$=$a {\sec}^{2} a x$
so as in the problem y = 10 tan 20x
the answer is $\frac{\mathrm{dy}}{\mathrm{dx}} = 10 \cdot 20 \cdot {\sec}^{2} \left(10 x\right) = 200 {\sec}^{2} \left(20 x\right)$