Question

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

Answer 1

1.) `y = (tan 3x)^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.) `dy/dx = 2tan 3x * sec^2 3x * 3`

The `2tan 3x` 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 3x` with respect to `x`, resulting in the `sec^2 3x`.

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

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

3.) `dy/dx = (6sin 3x)/(cos^3 3x)`