How do you find the derivative of y=tan^2(3x)?

Aug 1, 2014

1.) $y = {\left(\tan 3 x\right)}^{2}$

This is a problem that will involve a lot of chain rule. I will first show you what the derivative looks like and then explain where each part comes from:

2.) $\frac{\mathrm{dy}}{\mathrm{dx}} = 2 \tan 3 x \cdot {\sec}^{2} 3 x \cdot 3$

The $2 \tan 3 x$ is a result of first applying power rule. (bring the 2 out in front, and decrement the power)

Next, chain rule dictates that we multiply this with the derivative of the inside function $\tan 3 x$ with respect to $x$, resulting in the ${\sec}^{2} 3 x$.

And lastly, we apply chain rule again, multiplying the entire thing by $3$, which is the derivative of the $3 x$ inside the ${\sec}^{2} 3 x$.

The entire string can be prettified a bit by simplifying and rewriting in terms of $\sin$ and $\cos$:

3.) $\frac{\mathrm{dy}}{\mathrm{dx}} = \frac{6 \sin 3 x}{{\cos}^{3} 3 x}$