Chain rule and Derivatives . help. can't understand how solve.

Question

Chain rule and Derivatives of exponential and logarithmic functions

Answer 1

Power Rule: 1) h^'(s) = 4/5s^-1/5 - 2/3s^-1/3 = 4/(5 · ⁵√s) - 2/(3 · ³√s)

Chain Rule: 2) h(x) = log₃4 - 1/2·log₃ (x⁴ - x⁵)

h^'(x) = (5x⁴ - 4x³) / (2ln3·(x⁴ - x⁵))

3) y = ln(1 + e^x) - ln(1 - e^x)

y^' = e^x/(1 + e^x) + e^x/(1 - e^x) = 2e^x / (1 - e^2x)

Answer 2

I can give you some hints:

1) Use the power rule: d/ds sⁿ = ns^n-1 for both parts

2) Simplify first using log rules then take the derivative. Remember log_ax = ln(x)/ln(a)

h(x) = (1/ln(3))[ln(4)-2ln(x)-ln(1-x)/2]

Use d/dx ln(f(x)) = f'(x)/f(x)

To take the derivative of each part. Note: ln(4) is a constant

3) Simplify first using log division rule

y = ln(1+e^x)-ln(1-e^x)

Then take the derivative of each part using the chain rule. Then simplify the answer.