Calculus need help please

Question

2. Find the derivative for the function 

Answer 1

f(z) = z^5 + (e^z)(lnz)

f'(z) = 5z^4 +(e^z)(1/z) + (lnz)(e^z)

derivative = f'(z) = 5z^4 + e^z(1/z + lnz)

derivative of (e^z)(lnz) is the derivative of a product. It's the derivative of the 1st term (e^z) times the 2nd term, plus the derivative of the 2nd term (lnz) times the 1st term. [lnz(e^z)]' = lnz(e^z) + e^z(1/z)

derivative of 5z^4 follows the general rule that ax^b has derivative abx^(b-1)

so z^5 has derivative 5z^5-1) = 5z^4

e^z has itself as the derivative (e^z)' = e^z

lnz has 1/x as the derivative