§2.6 Derivatives of inverse functions

Introduction

Basic facts about inverse functions

The derivative of the natural logarithm function

The equation $y = \frac{5}{9}(x-32)$ converts Fahrenheit $x$ to Celsius $y$. Solving for $x$ gives $x = \frac{9}{5}y + 32$. Using this inverse, compute the Fahrenheit reading for $y = 20$ degrees Celsius.

For $h(x) = x^2\ln(x)$, which expression is $h'(x)$?

Which of the following equals $\frac{d}{dx}[\ln(x)]$ for $x \gt 0$?

For $s(y) = \ln(\cos(y) + 2)$, compute $s'(0)$.

For $z(x) = \tan(\ln(x))$, which expression is $z'(x)$?

For $m(z) = \ln(\ln(z))$, compute $m'(e)$ rounded to three decimals.

The natural logarithm is the inverse function of the natural exponential, which is why their derivatives are reciprocals of matching values.