3.8 HW Problems

Table of Contents:

Intro to inverse functions (video) | Khan Academy

Derivatives of inverse functions (video) | Khan Academy

• Remember that the natural logarithm function $f^{-1}(x)=\ln x$ as the inverse of the natural exponential function $f(x)=e^x$
  • this is one of the most important function-inverse pairs in mathematics and science

Derivatives of Inverses of Differentiable Functions

• Calculating the inverse of the function $f(x)=(\frac{1}{2})x+1$ we get $f^{-1}(x)=2x-2)$.
• If we calculate their derivatives, we see that:

$$ \frac{d}{dx}f(x)=\frac{d}{dx}\bigg(\frac{1}{2}x+1\bigg)=\frac{1}{2} \\ \frac{d}{dx}f^{-1}(x)=\frac{d}{dx}(2x-2)=2 $$

<aside> 💡 Graphing a line and its inverse together shows the graphs’ symmetry with respect to the line $y=x$. The slopes are reciprocals of each other.

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• Reflecting any nonhorizontal or nonvertical line across the line $y=x$ always inverts the line’s slope.
  • If the original line has slope $m \neq 0$, then the reflected line has a slope $\frac{1}{m}$.

Derivative of the Natural Logarithm Function