# What is the derivative of #y=arctan(cos(x))#?

##### 1 Answer

This is a case of knowing the how the derivative of inverse tangent works, and then following the chain rule.

If we were looking at

First remember that *x*," sometimes written as *x* and the *y* (among other things, but that's the important meaning here).

So

Notice the tangent is no longer an inverse after the switch.

Now we can use implicit differentiation. That's where we don't care whether or not we're looking at a function, that is, we don't care if we have *y* on one side and everything else on the other. We just derive everything as we go along, and we write *y* (that's a very, very over-simplified explanation of implicit differentiation, but it will work for this problem).

So we derive

and we get

which is the same as

Which seems straightforward enough. Except for one thing. We don't usually have derivatives that still have *y*'s in them. Not in the first year of Calculus, anyway. We should get rid of that *y*.

We can go back to a right triangle here. We started with *y* and the tangent of that angle is *x*, or

So:

Then by the Pythagorean Theorem, the hypotenuse can be found:

So the

meaning that if

then

which is the same as

So if we're looking at

The chain rule say that if you have an "inside function" and an "outside function," then you take the derivative of the outside function, and multiply that by the derivative of the inside function, or

If

Notice the inside function does not change when you derive the outside function.

If

Finally:

since the derivative of

Written more simply,

Hope this helps.