log(tan(2*x + 1)) ----------------- 4
log(tan(2*x + 1))/4
The derivative of a constant times a function is the constant times the derivative of the function.
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 :
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:
So, the result is:
Now simplify:
The answer is:
2 2 + 2*tan (2*x + 1) ------------------- 4*tan(2*x + 1)
2 / 2 \ 2 \1 + tan (1 + 2*x)/ 2 + 2*tan (1 + 2*x) - -------------------- 2 tan (1 + 2*x)
/ 2 \ | / 2 \ / 2 \| / 2 \ | \1 + tan (1 + 2*x)/ 2*\1 + tan (1 + 2*x)/| 4*\1 + tan (1 + 2*x)/*|2*tan(1 + 2*x) + -------------------- - ---------------------| | 3 tan(1 + 2*x) | \ tan (1 + 2*x) /