Differentiating Inverse Functions - AP Calculus BC

1. Find $(f^{-1})'(9)$ given $f(3)=9$ and $f'(3)=0$.

   Undefined because $f'(3)=0$. Cannot divide by zero when $f'(3)=0$.

2. What is the geometric meaning of inverse derivatives: how do the tangent line slopes compare at inverse points?

   Slopes are reciprocals at $(a,b)$ and $(b,a)$. Tangent lines at $(a,b)$ and $(b,a)$ have reciprocal slopes.

3. Find $(f^{-1})'(4)$ given $f(2)=4$ and $f'(2)=5$.

   $(f^{-1})'(4)=\frac{1}{5}$. Since $f(2)=4$ and $f'(2)=5$, use reciprocal formula.

4. Find $(f^{-1})'(2)$ if $f(x)=x^2+1$ with domain $[0,\infty)$.

   $(f^{-1})'(2)=\frac{1}{2}$. Since $f'(x)=2x$ and $f(1)=2$, so $f'(1)=2$, reciprocal is $\frac{1}{2}$.

5. Find $(f^{-1})'(e)$ if $f(x)=\ln(x)$.

   $(f^{-1})'(e)=e$. Since $e^x$ and $\ln(x)$ are inverses, $f'(1)=\frac{1}{e}$ gives reciprocal $e$.

6. Find $(f^{-1})'(\frac{\pi}{2})$ if $f(x)=\sin(x)$ with domain $\left[-\frac{\pi}{2},\frac{\pi}{2}\right]$.

   Undefined because $\cos\left(\frac{\pi}{2}\right)=0$. At $x=\frac{\pi}{2}$, $\sin'(x)=\cos(x)=0$, so undefined.

7. Find $(f^{-1})'(1)$ if $f(x)=e^x$.

   $(f^{-1})'(1)=1$. Since $\ln'(e)=\frac{1}{e}$ and $\ln(e)=1$, reciprocal is $e$.

8. Identify the graph transformation for inverses: across which line are $y=f(x)$ and $y=f^{-1}(x)$ reflected?

   Reflection across $y=x$. Inverse functions are mirror images across the diagonal line.

9. Identify the key condition needed to differentiate an inverse: what must be true about $f'(a)$?

   $f'(a)\ne 0$. Inverse derivative exists only when original derivative is nonzero.

10. Find $(f^{-1})'(1)$ given $f(0)=1$ and $f'(0)=\frac{1}{4}$.

    $(f^{-1})'(1)=4$. Since $f(0)=1$ and $f'(0)=\frac{1}{4}$, reciprocal is $4$.

11. What is the derivative of $y = \text{ln}|x|$?

    $\frac{1}{x}$ for $x \neq 0$. Chain rule applied to absolute value inside logarithm.

12. State the derivative formula for an inverse function: what is $(f^{-1})'(x)$ in terms of $f'$?

    $(f^{-1})'(x)=\frac{1}{f'(f^{-1}(x))}$. Derivative of inverse is reciprocal of original derivative at corresponding point.

13. Find $(f^{-1})'(0)$ if $f(x)=x+\sin(x)$ and $f(0)=0$.

    $(f^{-1})'(0)=\frac{1}{2}$. Since $f'(0)=1+\cos(0)=2$, reciprocal is $\frac{1}{2}$.

14. Find $(f^{-1})'(8)$ if $f(x)=x^3$ and $f^{-1}(8)=2$.

    $(f^{-1})'(8)=\frac{1}{12}$. Since $f'(x)=3x^2$ and $f'(2)=12$, reciprocal is $\frac{1}{12}$.

15. Identify the derivative of $y = \text{arctan}(x)$.

    $\frac{1}{1+x^2}$. Standard derivative formula for inverse tangent function.

16. What is the inverse-derivative relationship between slopes: how are $f'(a)$ and $(f^{-1})'(b)$ related when $f(a)=b$?

    $f'(a)\cdot (f^{-1})'(b)=1$. Slopes of inverse functions multiply to 1 at corresponding points.

17. What is the derivative of $y = \text{ln}(x)$?

    $\frac{1}{x}$. Standard derivative of natural logarithm function.

18. Find $(f^{-1})'(2)$ if $f(x)=x^3+1$ and $f^{-1}(2)=1$.

    $(f^{-1})'(2)=\frac{1}{3}$. Since $f'(x)=3x^2$ and $f'(1)=3$, reciprocal is $\frac{1}{3}$.

19. Which theorem relates derivatives of inverses to the original function?

    Inverse Function Theorem. States that $(f^{-1})'(x) = \frac{1}{f'(f^{-1}(x))}$ when $f'(f^{-1}(x)) \neq 0$.

20. Identify the correct expression: which equals $(f^{-1})'(x)$, $\frac{1}{f'(f^{-1}(x))}$ or $\frac{1}{f'(x)}$?

    $\frac{1}{f'(f^{-1}(x))}$. Must evaluate derivative at $f^{-1}(x)$, not at $x$ directly.

21. State the chain-rule identity used for inverse differentiation: what is $f(f^{-1}(x))$?

    $f(f^{-1}(x))=x$. Composing a function with its inverse yields the identity function.

22. State the derivative formula for the inverse function of $f(x)$.

    $\frac{1}{f'(f^{-1}(x))}$. Apply the Inverse Function Theorem: $(f^{-1})'(x) = \frac{1}{f'(f^{-1}(x))}$.

23. Identify the derivative of $y = \text{arccot}(x)$.

    $-\frac{1}{1+x^2}$. Derivative of inverse cotangent, negative of arctan derivative.

24. What is the equivalent point-slope form for inverses: if $f(a)=b$, what is $(f^{-1})'(b)$?

    $(f^{-1})'(b)=\frac{1}{f'(a)}$. When $f(a)=b$, slopes at $(a,b)$ and $(b,a)$ are reciprocals.

25. Find $(f^{-1})'(0)$ given $f(-3)=0$ and $f'(-3)=-2$.

    $(f^{-1})'(0)=-\frac{1}{2}$. Since $f(-3)=0$ and $f'(-3)=-2$, take reciprocal.

26. State the chain-rule identity used for inverse differentiation: what is $f^{-1}(f(x))$ (on the domain of invertibility)?

    $f^{-1}(f(x))=x$. Inverse followed by function returns original input on valid domain.

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