/ / / ___ \\\ tan\log\tan\\/ 3 *x - 1///
tan(log(tan(sqrt(3)*x - 1)))
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 :
Let .
The derivative of is .
Then, apply the chain rule. Multiply by :
Let .
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:
The result of the chain rule 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 :
Let .
The derivative of is .
Then, apply the chain rule. Multiply by :
Let .
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:
The result of the chain rule is:
The result of the chain rule is:
Now plug in to the quotient rule:
Now simplify:
The answer is:
___ / 2/ ___ \\ / 2/ / / ___ \\\\ \/ 3 *\1 + tan \\/ 3 *x - 1//*\1 + tan \log\tan\\/ 3 *x - 1//// --------------------------------------------------------------- / ___ \ tan\\/ 3 *x - 1/
/ 2/ ___\ / 2/ ___\\ / / / ___\\\\ / 2/ ___\\ / 2/ / / ___\\\\ | 1 + tan \-1 + x*\/ 3 / 2*\1 + tan \-1 + x*\/ 3 //*tan\log\tan\-1 + x*\/ 3 ///| 3*\1 + tan \-1 + x*\/ 3 //*\1 + tan \log\tan\-1 + x*\/ 3 ////*|2 - ---------------------- + ------------------------------------------------------| | 2/ ___\ 2/ ___\ | \ tan \-1 + x*\/ 3 / tan \-1 + x*\/ 3 / /
/ 2 2 2 2 \ | / 2/ ___\\ / 2/ ___\\ / 2/ ___\\ / 2/ / / ___\\\\ / 2/ ___\\ / / / ___\\\ / 2/ ___\\ 2/ / / ___\\\ / 2/ ___\\ / / / ___\\\| ___ / 2/ ___\\ / 2/ / / ___\\\\ | / ___\ \1 + tan \-1 + x*\/ 3 // 2*\1 + tan \-1 + x*\/ 3 // \1 + tan \-1 + x*\/ 3 // *\1 + tan \log\tan\-1 + x*\/ 3 //// 3*\1 + tan \-1 + x*\/ 3 // *tan\log\tan\-1 + x*\/ 3 /// 2*\1 + tan \-1 + x*\/ 3 // *tan \log\tan\-1 + x*\/ 3 /// 6*\1 + tan \-1 + x*\/ 3 //*tan\log\tan\-1 + x*\/ 3 ///| 6*\/ 3 *\1 + tan \-1 + x*\/ 3 //*\1 + tan \log\tan\-1 + x*\/ 3 ////*|2*tan\-1 + x*\/ 3 / + ------------------------- - -------------------------- + ------------------------------------------------------------ - ------------------------------------------------------- + -------------------------------------------------------- + ------------------------------------------------------| | 3/ ___\ / ___\ 3/ ___\ 3/ ___\ 3/ ___\ / ___\ | \ tan \-1 + x*\/ 3 / tan\-1 + x*\/ 3 / tan \-1 + x*\/ 3 / tan \-1 + x*\/ 3 / tan \-1 + x*\/ 3 / tan\-1 + x*\/ 3 / /