log(tan(x))
d --(log(tan(x))) dx
Let .
The derivative of is .
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:
Now simplify:
The answer is:
2 1 + tan (x) ----------- tan(x)
2 / 2 \ 2 \1 + tan (x)/ 2 + 2*tan (x) - -------------- 2 tan (x)
/ 2 \ | / 2 \ / 2 \| / 2 \ | \1 + tan (x)/ 2*\1 + tan (x)/| 2*\1 + tan (x)/*|2*tan(x) + -------------- - ---------------| | 3 tan(x) | \ tan (x) /