___ cot(x) - \/ x
d / ___\ --\cot(x) - \/ x / dx
Differentiate term by term:
There are multiple ways to do this derivative.
Rewrite the function to be differentiated:
Let .
Apply the power rule: goes to
Then, apply the chain rule. Multiply by :
Rewrite the function to be differentiated:
Apply the quotient rule, which is:
and .
To find :
The derivative of sine is cosine:
To find :
The derivative of cosine is negative sine:
Now plug in to the quotient rule:
The result of the chain rule is:
Rewrite the function to be differentiated:
Apply the quotient rule, which is:
and .
To find :
The derivative of cosine is negative sine:
To find :
The derivative of sine is cosine:
Now plug in to the quotient rule:
The derivative of a constant times a function is the constant times the derivative of the function.
Apply the power rule: goes to
So, the result is:
The result is:
Now simplify:
The answer is:
2 1 -1 - cot (x) - ------- ___ 2*\/ x
1 / 2 \ ------ + 2*\1 + cot (x)/*cot(x) 3/2 4*x
/ 2 \ | / 2 \ 3 2 / 2 \| -|2*\1 + cot (x)/ + ------ + 4*cot (x)*\1 + cot (x)/| | 5/2 | \ 8*x /