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
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
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
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.