Question

find deriv of f(x) =cot^3 (sq root x)
plz show work and ty dont wanna fail my exam tmmrw D:

Answer 1

\[\huge f(x)=\cot^3 \sqrt x\]Like this? :o

Answer 2

So we'll work from the outside inward.
We differentiate the outermost function first.

In this case, the outermost function is the CUBE power,
After we differentiate the outside, we make a copy of the inside, and multiply by it's derivative.

\[\huge f'(x)=3\cot^2 \sqrt x(\color{royalblue}{\cot \sqrt x})'\]
The prime is to show that we still need to take a derivative of this part.

Answer 3

Taking the derivative of the blue part gives us,\[\huge f'(x)=3\cot^2 \sqrt x(\color{royalblue}{-\csc^2 \sqrt x})(\color{forestgreen}{\sqrt x})'\]

Answer 4

We apply the chain rule again, multiplying by the derivative of the inner inner function, sqrt x.

Answer 5

Taking the derivative of the green part gives us,\[\huge f'(x)=3\cot^2 \sqrt x(\color{royalblue}{-\csc^2 \sqrt x})\left(\color{forestgreen}{\frac{1}{2\sqrt x}}\right)\]

Answer 6

Confused on any part of that? I know it's a ton of steps.

If you can understand the chain rule ok, then it's a little easier to get through.