/ / 2 \\ log\sin\x + 1//
d / / / 2 \\\ --\log\sin\x + 1/// dx
Let .
The derivative of is .
Then, apply the chain rule. Multiply by :
Let .
The derivative of sine is cosine:
Then, apply the chain rule. Multiply by :
Differentiate term by term:
Apply the power rule: goes to
The derivative of the constant is zero.
The result is:
The result of the chain rule is:
The result of the chain rule is:
Now simplify:
The answer is:
/ 2 \ 2*x*cos\x + 1/ --------------- / 2 \ sin\x + 1/
/ / 2\ 2 2/ 2\\ | 2 cos\1 + x / 2*x *cos \1 + x /| 2*|- 2*x + ----------- - -----------------| | / 2\ 2/ 2\ | \ sin\1 + x / sin \1 + x / /
/ 2/ 2\ 2 3/ 2\ 2 / 2\\ | 3*cos \1 + x / 4*x *cos \1 + x / 4*x *cos\1 + x /| 4*x*|-3 - -------------- + ----------------- + ----------------| | 2/ 2\ 3/ 2\ / 2\ | \ sin \1 + x / sin \1 + x / sin\1 + x / /