/ / 2\\ log\tan\3*x //
log(tan(3*x^2))
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/ 2\\ 6*x*\1 + tan \3*x // -------------------- / 2\ tan\3*x /
/ 2 / 2/ 2\\\ / 2/ 2\\ | 1 2 6*x *\1 + tan \3*x //| 6*\1 + tan \3*x //*|--------- + 12*x - ---------------------| | / 2\ 2/ 2\ | \tan\3*x / tan \3*x / /
/ 2\ | 2/ 2\ 2 / 2/ 2\\ 2 / 2/ 2\\ | / 2/ 2\\ | 1 + tan \3*x / 2 / 2\ 8*x *\1 + tan \3*x // 4*x *\1 + tan \3*x // | 108*x*\1 + tan \3*x //*|2 - -------------- + 8*x *tan\3*x / - --------------------- + ----------------------| | 2/ 2\ / 2\ 3/ 2\ | \ tan \3*x / tan\3*x / tan \3*x / /