log(x*sin(x) + cos(x))
log(x*sin(x) + cos(x))
Let .
The derivative of is .
Then, apply the chain rule. Multiply by :
Differentiate term by term:
Apply the product rule:
; to find :
Apply the power rule: goes to
; to find :
The derivative of sine is cosine:
The result is:
The derivative of cosine is negative sine:
The result is:
The result of the chain rule is:
The answer is:
x*cos(x) ----------------- x*sin(x) + cos(x)
2 2 x *cos (x) -x*sin(x) - ----------------- + cos(x) x*sin(x) + cos(x) -------------------------------------- x*sin(x) + cos(x)
2 3 3 2 3*x*cos (x) 2*x *cos (x) 3*x *cos(x)*sin(x) -2*sin(x) - x*cos(x) - ----------------- + -------------------- + ------------------ x*sin(x) + cos(x) 2 x*sin(x) + cos(x) (x*sin(x) + cos(x)) ------------------------------------------------------------------------------------ x*sin(x) + cos(x)