Differentiate the function.

Question

H(z) = ln(a²-z²/a²+z²

Answer 1

First simplify using properties of logs:

H(z) = (1/2)ln[(a² - z²) / (a² + z²)] = (1/2)[ln(a² - z²) - ln(a² + z²)]

So, H'(z) = (1/2)[-2z / (a² - z²) - 2z / (a² + z²) = -z[1 / (a² - z²) + 1 / (a² + z²)] =

-z[(a² + z²) + (a² - z²)] / (a⁴ - z⁴) = -2a²z / (a⁴ - z⁴)

Answer 2

1. Use log rules to re-write H(z)
2. 1/2 ln (a²-z²/a²⁺z²)
3. 1/2 (ln(a²-z²)-ln(a²+z²))
4. Differentiate
5. 1/2 (-2z/(a²-z²) - 2z/(a²⁺z²))
6. (-z(a²⁺z²)-z(a²-z²))/(a⁴-z⁴)
7. Simplify
8. -2a²z/a⁴-z⁴