__________ cos(3*x) \/ sin(5*x) *E
sqrt(sin(5*x))*E^cos(3*x)
Apply the product rule:
; to find :
Let .
Apply the power rule: goes to
Then, apply the chain rule. Multiply by :
Let .
The derivative of sine is cosine:
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 of the chain rule is:
; to find :
Let .
The derivative of is itself.
Then, apply the chain rule. Multiply by :
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 of the chain rule is:
The result is:
Now simplify:
The answer is:
cos(3*x) __________ cos(3*x) 5*cos(5*x)*e - 3*\/ sin(5*x) *e *sin(3*x) + -------------------- __________ 2*\/ sin(5*x)
/ __________ 2 \ | 25*\/ sin(5*x) __________ / 2 \ 25*cos (5*x) 15*cos(5*x)*sin(3*x)| cos(3*x) |- --------------- + 9*\/ sin(5*x) *\sin (3*x) - cos(3*x)/ - ------------- - --------------------|*e | 2 3/2 __________ | \ 4*sin (5*x) \/ sin(5*x) /
/ / 2 \ / 2 \ \ | | __________ cos (5*x) | | 3*cos (5*x)| | |225*|2*\/ sin(5*x) + -----------|*sin(3*x) 125*|2 + -----------|*cos(5*x) | | | 3/2 | | 2 | / 2 \ | | \ sin (5*x)/ __________ / 2 \ \ sin (5*x) / 135*\sin (3*x) - cos(3*x)/*cos(5*x)| cos(3*x) |------------------------------------------- + 27*\/ sin(5*x) *\1 - sin (3*x) + 3*cos(3*x)/*sin(3*x) + ------------------------------ + -----------------------------------|*e | 4 __________ __________ | \ 8*\/ sin(5*x) 2*\/ sin(5*x) /