Question

Hi, I have a calculus test tomorrow and I understand how to complete all of the problems, but I am unsure behind the reasoning of taking two derivatives. Can someone please explain the following:

1. g(x)=e^(f(x))
g'(x)=e^(f(x))f'(x)

2. f(x)=Integral from 0 to 3x of the square root of (4+t^2)dt
f'(x) = 3 times the square root of (4+9x^2).

For the second derivative, I know it has something to do with the 2nd fundamental theorem of calculus, but if you could elaborate I would appreciate it greatly.

Thank you very much.

Answer 1

1) d/dx e^(u) is always (u')(e^(u)). In other words, take the original function and multiply it by the derivative of u (f(x)).

2) The 2nd FTC: Take the b (top number of the integral) and plug that in for t. Multiply by the derivative of b. b'(f(b)).