log(tan(2*x) + 1/4)
log(tan(2*x) + 1/4)
Let .
The derivative of is .
Then, apply the chain rule. Multiply by :
Differentiate term by term:
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 derivative of the constant is zero.
The result is:
The result of the chain rule is:
Now simplify:
The answer is:
2 2 + 2*tan (2*x) --------------- tan(2*x) + 1/4
/ / 2 \ \ / 2 \ | 2*\1 + tan (2*x)/ | 32*\1 + tan (2*x)/*|- ----------------- + tan(2*x)| \ 1 + 4*tan(2*x) / --------------------------------------------------- 1 + 4*tan(2*x)
/ 2 \ | / 2 \ / 2 \ | / 2 \ | 2 16*\1 + tan (2*x)/ 12*\1 + tan (2*x)/*tan(2*x)| 64*\1 + tan (2*x)/*|1 + 3*tan (2*x) + ------------------- - ---------------------------| | 2 1 + 4*tan(2*x) | \ (1 + 4*tan(2*x)) / ---------------------------------------------------------------------------------------- 1 + 4*tan(2*x)