/ 3 \ \5*x + 4*x + 2/*cot(2*x)
d // 3 \ \ --\\5*x + 4*x + 2/*cot(2*x)/ dx
Apply the product rule:
; to find :
Differentiate term by term:
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 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 derivative of the constant is zero.
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:
/ 2 \ / 3 \ / 2\ \-2 - 2*cot (2*x)/*\5*x + 4*x + 2/ + \4 + 15*x /*cot(2*x)
/ / 2 \ / 2\ / 2 \ / 3\ \ 2*\- 2*\1 + cot (2*x)/*\4 + 15*x / + 15*x*cot(2*x) + 4*\1 + cot (2*x)/*\2 + 4*x + 5*x /*cot(2*x)/
/ / 2 \ / 2 \ / 2 \ / 3\ / 2 \ / 2\ \ 2*\15*cot(2*x) - 90*x*\1 + cot (2*x)/ - 8*\1 + cot (2*x)/*\1 + 3*cot (2*x)/*\2 + 4*x + 5*x / + 12*\1 + cot (2*x)/*\4 + 15*x /*cot(2*x)/