Chapter 14.2, Problem 31P

Problem 28 is the chain rule for the derivative of a function of a function. Hint: Assume that $df/dg$ and $dg/dz$ exist, and write equations like (3.5) of Chapter 4 for $\Delta f$ and $\Delta g$ substitute $\Delta g$ into $\Delta f,$ divide by $\Delta z,$ and take limits.

$\frac{d}{dz}\ln z=\frac{1}{z},z\ne 0.$ Hint: Expand $\ln \left(1+\frac{\Delta z}{z}\right)$ in series.