1 cot(x) 1*------- + ------ 2 3 cot (x)
d / 1 cot(x)\ --|1*------- + ------| dx| 2 3 | \ cot (x) /
Differentiate term by term:
Apply the quotient rule, which is:
and .
To find :
The derivative of the constant is zero.
To find :
Let .
Apply the power rule: goes to
Then, apply the chain rule. Multiply by :
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 result of the chain rule is:
Now plug in to the quotient rule:
The derivative of a constant times a function is the constant times the derivative of the function.
So, the result is:
The result is:
Now simplify:
The answer is:
2 2 1 cot (x) -2 - 2*cot (x) - - - ------- - -------------- 3 3 2 cot(x)*cot (x)
/ 2 \ / / 2 \ \ |1 cot (x)| | 6 9*\1 + cot (x)/ | 2*|- + -------|*|- ------- + --------------- + cot(x)| \3 3 / | 2 4 | \ cot (x) cot (x) /
/ 2\ / 2 \ | / 2 \ / 2 \ | |1 cot (x)| | 2 12 48*\1 + cot (x)/ 36*\1 + cot (x)/ | 2*|- + -------|*|-1 - 3*cot (x) + ------ - ---------------- + -----------------| \3 3 / | cot(x) 3 5 | \ cot (x) cot (x) /