cot(3*x) --------*cos(5*x) 4
(cot(3*x)/4)*cos(5*x)
Apply the quotient rule, which is:
and .
To find :
Apply the product rule:
; 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 :
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:
To find :
The derivative of the constant is zero.
Now plug in to the quotient rule:
Now simplify:
The answer is:
/ 2 \ | 3 3*cot (3*x)| 5*cot(3*x)*sin(5*x) |- - - -----------|*cos(5*x) - ------------------- \ 4 4 / 4
/ 2 \ / 2 \ -25*cos(5*x)*cot(3*x) + 30*\1 + cot (3*x)/*sin(5*x) + 18*\1 + cot (3*x)/*cos(5*x)*cot(3*x) ------------------------------------------------------------------------------------------ 4
/ 2 \ / 2 \ / 2 \ / 2 \ 125*cot(3*x)*sin(5*x) + 225*\1 + cot (3*x)/*cos(5*x) - 270*\1 + cot (3*x)/*cot(3*x)*sin(5*x) - 54*\1 + cot (3*x)/*\1 + 3*cot (3*x)/*cos(5*x) -------------------------------------------------------------------------------------------------------------------------------------------- 4