/ / cos(x)\\ log|sin|x - ------|| \ \ 4 //
log(sin(x - cos(x)/4))
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 a constant times a function is the constant times the derivative of the function.
The derivative of cosine is negative sine:
So, the result is:
The result is:
The result of the chain rule is:
The result of the chain rule is:
Now simplify:
The answer is:
/ sin(x)\ / cos(x)\ |1 + ------|*cos|x - ------| \ 4 / \ 4 / ---------------------------- / cos(x)\ sin|x - ------| \ 4 /
2 2/ cos(x)\ / cos(x)\ (4 + sin(x)) *cos |x - ------| 4*cos(x)*cos|x - ------| 2 \ 4 / \ 4 / - (4 + sin(x)) - ------------------------------ + ------------------------ 2/ cos(x)\ / cos(x)\ sin |x - ------| sin|x - ------| \ 4 / \ 4 / --------------------------------------------------------------------------- 16
3 3/ cos(x)\ 3 / cos(x)\ / cos(x)\ 2/ cos(x)\ (4 + sin(x)) *cos |x - ------| (4 + sin(x)) *cos|x - ------| 8*cos|x - ------|*sin(x) 6*cos |x - ------|*(4 + sin(x))*cos(x) \ 4 / \ 4 / \ 4 / \ 4 / -6*(4 + sin(x))*cos(x) + ------------------------------ + ----------------------------- - ------------------------ - -------------------------------------- 3/ cos(x)\ / cos(x)\ / cos(x)\ 2/ cos(x)\ sin |x - ------| sin|x - ------| sin|x - ------| sin |x - ------| \ 4 / \ 4 / \ 4 / \ 4 / ----------------------------------------------------------------------------------------------------------------------------------------------------------- 32