/ 2 1\ log|tan (x) + -| \ 4/
log(tan(x)^2 + 1/4)
Let .
The derivative of is .
Then, apply the chain rule. Multiply by :
Differentiate term by term:
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 :
The derivative of sine is cosine:
To find :
The derivative of cosine is negative sine:
Now plug in to the quotient rule:
The result of the chain rule is:
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 (x)/*tan(x) ---------------------- 2 1 tan (x) + - 4
/ 2 / 2 \\ / 2 \ | 2 8*tan (x)*\1 + tan (x)/| 8*\1 + tan (x)/*|1 + 3*tan (x) - -----------------------| | 2 | \ 1 + 4*tan (x) / --------------------------------------------------------- 2 1 + 4*tan (x)
/ 2 2 \ | / 2 \ 2 / 2 \ / 2 \ 2 | / 2 \ | 2 6*\1 + tan (x)/ 12*tan (x)*\1 + tan (x)/ 32*\1 + tan (x)/ *tan (x)| 32*\1 + tan (x)/*|2 + 3*tan (x) - ---------------- - ------------------------ + -------------------------|*tan(x) | 2 2 2 | | 1 + 4*tan (x) 1 + 4*tan (x) / 2 \ | \ \1 + 4*tan (x)/ / ----------------------------------------------------------------------------------------------------------------- 2 1 + 4*tan (x)