/ 3\ log\tan(3*x) + x /
d / / 3\\ --\log\tan(3*x) + x // dx
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:
Apply the power rule: goes to
The result is:
The result of the chain rule is:
Now simplify:
The answer is:
2 2 3 + 3*x + 3*tan (3*x) ---------------------- 3 tan(3*x) + x
/ 2 \ | / 2 2 \ | | 3*\1 + x + tan (3*x)/ / 2 \ | 3*|2*x - ----------------------- + 6*\1 + tan (3*x)/*tan(3*x)| | 3 | \ x + tan(3*x) / -------------------------------------------------------------- 3 x + tan(3*x)
/ 3 \ | 2 / 2 2 \ / / 2 \ \ / 2 2 \| | / 2 \ 9*\1 + x + tan (3*x)/ 2 / 2 \ 9*\x + 3*\1 + tan (3*x)/*tan(3*x)/*\1 + x + tan (3*x)/| 6*|1 + 9*\1 + tan (3*x)/ + ----------------------- + 18*tan (3*x)*\1 + tan (3*x)/ - -------------------------------------------------------| | 2 3 | | / 3 \ x + tan(3*x) | \ \x + tan(3*x)/ / --------------------------------------------------------------------------------------------------------------------------------------------- 3 x + tan(3*x)