2 cos (x) x*cos(x)*sin(x) + ------- 2
(x*cos(x))*sin(x) + cos(x)^2/2
Differentiate term by term:
Apply the product rule:
; to find :
Apply the product rule:
; to find :
Apply the power rule: goes to
; to find :
The derivative of cosine is negative sine:
The result is:
; to find :
The derivative of sine is cosine:
The result is:
The derivative of a constant times a function is the constant times the derivative of the function.
Let .
Apply the power rule: goes to
Then, apply the chain rule. Multiply by :
The derivative of cosine is negative sine:
The result of the chain rule is:
So, the result is:
The result is:
Now simplify:
The answer is:
2 x*cos (x) + (-x*sin(x) + cos(x))*sin(x) - cos(x)*sin(x)
2 sin (x) - (-cos(x) + x*sin(x))*cos(x) - (2*sin(x) + x*cos(x))*sin(x) - 2*x*cos(x)*sin(x)
2 2 (-cos(x) + x*sin(x))*sin(x) + (-3*cos(x) + x*sin(x))*sin(x) - 2*x*cos (x) - 2*(2*sin(x) + x*cos(x))*cos(x) + 2*x*sin (x)