Apply the product rule:
; to find :
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:
; 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.
Apply the power rule: goes to
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.
Apply the power rule: goes to
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.
Apply the power rule: goes to
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.
Apply the power rule: goes to
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:
8 / 2 \ 7 6*x *\-5 - 5*cot (5*x)/ + 48*x *cot(5*x)
6 / / 2 \ 2 / 2 \ \ 12*x *\28*cot(5*x) - 40*x*\1 + cot (5*x)/ + 25*x *\1 + cot (5*x)/*cot(5*x)/
5 / / 2 \ 3 / 2 \ / 2 \ 2 / 2 \ \ 12*x *\168*cot(5*x) - 420*x*\1 + cot (5*x)/ - 125*x *\1 + cot (5*x)/*\1 + 3*cot (5*x)/ + 600*x *\1 + cot (5*x)/*cot(5*x)/