Question

What is the derivative of `y= sin(tan 2x)`?

Answer 1

`dy/dx = 2cos(tan2x)sec^2(2x)`

Explanation:

We need to apply the chain rule twice.

Recall that the chain rule states, if we have some function `f(g(x))`, the derivative of `f` with respect to `x` is equal to the derivative of `f` with respect to `g`, multiplied by the derivative of `g` with respect to `x`.

So in this case, the derivative `dy/dx` will equal the derivative of `sin(tan 2x)` with respect to `tan 2x` (basically, treat `tan 2x` as a whole variable) times the derivative of `tan 2x` with respect to `x`.

Derivative of `sin` is just `cos`:

`dy/dx = cos(tan 2x) * d/dx[tan 2x]`

Derivative of `tan` is `sec^2`. However, we need to apply the chain rule again, meaning this time we will just pull the derivative of `2x` out. (which is just `2`)

`dy/dx = cos(tan 2x) * sec^2(2x) * 2`