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 :
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 the constant is zero.
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 :
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 the constant is zero.
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 :
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 the constant is zero.
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 :
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 the constant is zero.
The result is:
The result of the chain rule is:
Now plug in to the quotient rule:
Now simplify:
The answer is:
/ 2/ 2 \\ 12*x*\-1 - cot \6*x - 9//
/ 2/ / 2\\ 2 / 2/ / 2\\\ / / 2\\\ 12*\-1 - cot \3*\-3 + 2*x // + 24*x *\1 + cot \3*\-3 + 2*x ///*cot\3*\-3 + 2*x ///
/ 2/ / 2\\\ / 2 2/ / 2\\ 2 / 2/ / 2\\\ / / 2\\\ 864*x*\1 + cot \3*\-3 + 2*x ///*\- 8*x *cot \3*\-3 + 2*x // - 4*x *\1 + cot \3*\-3 + 2*x /// + cot\3*\-3 + 2*x ///