___________ / / 2\ \/ tan\3*x /
/ ___________\ d | / / 2\ | --\\/ tan\3*x / / dx
Let .
Apply the power rule: goes to
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\\ 3*x*\1 + tan \3*x // -------------------- ___________ / / 2\ \/ tan\3*x /
/ ___________ 2 / 2/ 2\\\ / 2/ 2\\ | 1 2 / / 2\ 3*x *\1 + tan \3*x //| 3*\1 + tan \3*x //*|-------------- + 12*x *\/ tan\3*x / - ---------------------| | ___________ 3/2/ 2\ | | / / 2\ tan \3*x / | \\/ tan\3*x / /
/ 2\ | ___________ 2/ 2\ 2 / 2/ 2\\ 2 / 2/ 2\\ | / 2/ 2\\ | / / 2\ 1 + tan \3*x / 2 3/2/ 2\ 4*x *\1 + tan \3*x // 3*x *\1 + tan \3*x // | 27*x*\1 + tan \3*x //*|4*\/ tan\3*x / - -------------- + 16*x *tan \3*x / - --------------------- + ----------------------| | 3/2/ 2\ ___________ 5/2/ 2\ | | tan \3*x / / / 2\ tan \3*x / | \ \/ tan\3*x / /