log(tan(4*x))
d --(log(tan(4*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 :
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:
Now simplify:
The answer is:
2 4 + 4*tan (4*x) --------------- tan(4*x)
/ 2\ | / 2 \ | | 2 \1 + tan (4*x)/ | 16*|2 + 2*tan (4*x) - ----------------| | 2 | \ tan (4*x) /
/ 2 \ | / 2 \ / 2 \| / 2 \ | \1 + tan (4*x)/ 2*\1 + tan (4*x)/| 128*\1 + tan (4*x)/*|2*tan(4*x) + ---------------- - -----------------| | 3 tan(4*x) | \ tan (4*x) /