Question

What is the derivative of `cos(tan x)`?

Answer 1

`-sin(tan(x))sec^2(x)`

Explanation:

We can use the chain rule to create a rule for differentiating a cosine function:

Since `d/dx(cos(x))=-sin(x)`, we know that `d/dx(cos(u))=-sin(u)*u'`.

Applying this to `cos(tan(x))`, where `u=tan(x)`, we see that

`d/dx(cos(tan(x)))=-sin(tan(x))*d/dx(tan(x))`

It is helpful to know that `d/dx(tan(x))=sec^2(x)`. Hence

`d/dx(cos(tan(x)))=-sin(tan(x))sec^2(x)`