# How do you take the derivative of tan^2(5x)?

Aug 6, 2015

$= 10 \tan 5 x {\sec}^{2} 5 x$

#### Explanation:

Let $t = 5 x$.

$\frac{d}{\mathrm{dx}} {\tan}^{2} 5 x = \frac{\mathrm{dt}}{\mathrm{dx}} \frac{d}{\mathrm{dt}} {\tan}^{2} t$.

Product rule: $\frac{d}{\mathrm{dt}} {\tan}^{2} t = 2 \tan t \setminus \frac{d}{\mathrm{dt}} \tan t$.
Now $\setminus \frac{d}{\mathrm{dt}} \tan t = {\sec}^{2} t$.

Hence $\frac{d}{\mathrm{dx}} {\tan}^{2} 5 x = 10 \tan 5 x {\sec}^{2} 5 x$