M171 Calculus I · Unit 2: Computing derivatives · on Exam 2
§2.6 Derivatives of inverse functions
Study notes for Derivatives of inverse functions in M171 Calculus I at Montana State University: the big ideas, key formulas, a worked example, common mistakes and an exam tip. The same topic has endless practice questions and flashcards in Mathub.
Big ideas
- If $f$ has inverse $f^{-1}$, then $(f^{-1})'(x) = \dfrac{1}{f'\big(f^{-1}(x)\big)}$. The slope of the inverse at $(b,a)$ is the reciprocal of the slope of $f$ at $(a,b)$.
- This produces the derivatives of $\ln x$ (inverse of $e^x$), $\log_b x$, $\arcsin x$, $\arctan x$, $\arccos x$.
- With the chain rule: $\frac{d}{dx}[\ln(u)] = \frac{u'}{u}$, $\frac{d}{dx}[\arctan(u)] = \frac{u'}{1+u^2}$.
Key formulas
Inverse function rule
$$\big(f^{-1}\big)'(x) = \frac{1}{f'\!\left(f^{-1}(x)\right)}$$
Logarithms
$$\frac{d}{dx}[\ln x] = \frac1x,\qquad \frac{d}{dx}[\log_b x] = \frac{1}{x\ln b}$$
Inverse trig
$$\frac{d}{dx}[\arcsin x] = \frac{1}{\sqrt{1-x^2}},\quad \frac{d}{dx}[\arctan x] = \frac{1}{1+x^2},\quad \frac{d}{dx}[\arccos x] = -\frac{1}{\sqrt{1-x^2}}$$
Worked example
Given $f(2) = 5$ and $f'(2) = 3$, find $(f^{-1})'(5)$. Also differentiate $y = \ln(x^2+1)$.
Show the solution
$(f^{-1})'(5) = \dfrac{1}{f'(f^{-1}(5))} = \dfrac{1}{f'(2)} = \dfrac13$. For $y$: $y' = \dfrac{2x}{x^2+1}$.
Common mistakes
- Computing $(f^{-1})'(5)$ as $1/f'(5)$; you must evaluate $f'$ at $f^{-1}(5)$, not at $5$.
- Writing $\frac{d}{dx}[\ln(x^2+1)] = \frac{1}{x^2+1}$ without the chain factor $2x$.
Exam tipThink of $\ln$ as "the exponent": $\ln(x^2+1)$ asks what power of $e$ gives $x^2+1$, so its derivative is inside$'$ over inside.
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