/x*pi\ sin(2*x)*cot|----| \ 4 /
sin(2*x)*cot((x*pi)/4)
Apply the product rule:
; to find :
Let .
The derivative of sine is cosine:
Then, apply the chain rule. Multiply by :
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 of the chain rule is:
; to find :
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 :
Let .
The derivative of sine is cosine:
Then, apply the chain rule. Multiply by :
The derivative of a constant times a function is the constant times the derivative of the function.
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:
So, the result is:
The result of the chain rule is:
To find :
Let .
The derivative of cosine is negative sine:
Then, apply the chain rule. Multiply by :
The derivative of a constant times a function is the constant times the derivative of the function.
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:
So, the result is:
The result of the chain rule is:
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 :
Let .
The derivative of cosine is negative sine:
Then, apply the chain rule. Multiply by :
The derivative of a constant times a function is the constant times the derivative of the function.
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:
So, the result is:
The result of the chain rule is:
To find :
Let .
The derivative of sine is cosine:
Then, apply the chain rule. Multiply by :
The derivative of a constant times a function is the constant times the derivative of the function.
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:
So, the result is:
The result of the chain rule is:
Now plug in to the quotient rule:
The result is:
Now simplify:
The answer is:
/ 2/x*pi\\ pi*|-1 - cot |----||*sin(2*x) /x*pi\ \ \ 4 // 2*cos(2*x)*cot|----| + ----------------------------- \ 4 / 4
2 / 2/pi*x\\ /pi*x\ pi *|1 + cot |----||*cot|----|*sin(2*x) /pi*x\ / 2/pi*x\\ \ \ 4 // \ 4 / - 4*cot|----|*sin(2*x) - pi*|1 + cot |----||*cos(2*x) + --------------------------------------- \ 4 / \ \ 4 // 8
3 / 2/pi*x\\ / 2/pi*x\\ 2 / 2/pi*x\\ /pi*x\ pi *|1 + cot |----||*|1 + 3*cot |----||*sin(2*x) 3*pi *|1 + cot |----||*cos(2*x)*cot|----| /pi*x\ / 2/pi*x\\ \ \ 4 // \ \ 4 // \ \ 4 // \ 4 / - 8*cos(2*x)*cot|----| + 3*pi*|1 + cot |----||*sin(2*x) - ------------------------------------------------ + ----------------------------------------- \ 4 / \ \ 4 // 32 4