x*(sin(log(x)) - cos(log(x)))
x*(sin(log(x)) - cos(log(x)))
Apply the product rule:
; to find :
Apply the power rule: goes to
; to find :
Differentiate term by term:
Let .
The derivative of sine is cosine:
Then, apply the chain rule. Multiply by :
The derivative of is .
The result of the chain rule is:
The derivative of a constant times a function is the constant times the derivative of the function.
Let .
The derivative of cosine is negative sine:
Then, apply the chain rule. Multiply by :
The derivative of is .
The result of the chain rule is:
So, the result is:
The result is:
The result is:
Now simplify:
The answer is:
/cos(log(x)) sin(log(x))\ -cos(log(x)) + x*|----------- + -----------| + sin(log(x)) \ x x /
2*cos(log(x)) ------------- x
2*(-cos(log(x)) - sin(log(x))) ------------------------------ 2 x