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