To find the derivative of?

Question

f(x)= cos (e^x) +e ^x sin (e^x)
-e^x sin (e^x) + e^x sin (e^x) + e^2x cos (e^x)
The part I don’t get is why is e^x sin(e^x) there
Isnt it just : -e^x sin (e^x) + e^2x cos (e^x)

yea i get the way it supposed to be but can u tell me why its like that...is there a rule or something

Answer 1

f(x)= cos (e^x) +e ^x sin (e^x)
f'(x)=-sin(e^x)*(e^x)+e^x*sin(e^x)+e^x*cos(e^x)*e^x
f'(x)=e^x(-sin(e^x)+sin(e^x)+e^x*cos(e^x))
f'(x)=e^x(e^x*cos(e^x))

Answer 2

Looks like you're happy with d/dx cos e^x = -e^x sin e^x, so I'll leave those terms out.

From the product rule:
d/dx (e^x sin (e^x)) = e^x d/dx (sin e^x) + (d/dx e^x) sin e^x
= e^x [cos e^x . e^x] + e^x sin e^x
= e^2x cos e^x + e^x sin e^x.