3/8 4 x *cot(5*x) - -- 5 x
d / 3/8 4 \ --|x *cot(5*x) - --| dx| 5| \ x /
Differentiate term by term:
Apply the product rule:
; to find :
Apply the power rule: goes to
; 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:
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 is:
Now simplify:
The answer is:
20 3/8 / 2 \ 3*cot(5*x) -- + x *\-5 - 5*cot (5*x)/ + ---------- 6 5/8 x 8*x
/ / 2 \ \ | 24 3*\1 + cot (5*x)/ 3*cot(5*x) 3/8 / 2 \ | 5*|- -- - ----------------- - ---------- + 10*x *\1 + cot (5*x)/*cot(5*x)| | 7 5/8 13/8 | \ x 4*x 64*x /
/ 2 / 2 \ / 2 \ \ |168 3/8 / 2 \ 39*cot(5*x) 45*\1 + cot (5*x)/ 3/8 2 / 2 \ 45*\1 + cot (5*x)/*cot(5*x)| 5*|--- - 50*x *\1 + cot (5*x)/ + ----------- + ------------------ - 100*x *cot (5*x)*\1 + cot (5*x)/ + ---------------------------| | 8 21/8 13/8 5/8 | \ x 512*x 64*x 4*x /