Please Help! solve the following questions based on the derivative of exponential functions.

Question

Solve the following questions based on the derivative of exponential functions.

1. a) If f(x) = 1/3 (e^3x + e^-3x), calculate f'(1).

b) If f(x) = e^{-(1/x+1) (1 over all of x+1)}, calculate f'(0).

c) If h(z) = z²(1 + e^-z), calculate h^'(-1).

Answer 1

a) f'(x) = (3e^3x-3e^-3x)/3 = e^3x-e^-3x

f'(1) = e³ - e^-3

b) f'(x) = e^-1/(x+1)(1/(x+1)²)

f'(0) = e^-1/4

c)h'(z) = 2z(1+e^-z)+z²(-e^-z)

h'(-1) = 2(-1)(1+e) -e^-1 = -2-2e-e = -2-3e