- Find the derivative of an inverse function
- Identify the derivatives for inverse trig functions like arcsine, arccosine, and arctangent
Derivatives of Various Inverse Functions
The Main Idea
- Inverse Function Theorem:
- For invertible and differentiable function f(x): [latex]\dfrac{1}{f^{\prime}(f^{-1}(x))}[/latex]
- Graphical Interpretation:
- Tangent lines of [latex]f(x)[/latex] and [latex]f^(-1)(x)[/latex] have reciprocal slopes
- Symmetric about [latex]y = x[/latex] line
- Extending the Power Rule:
- For positive integer [latex]n[/latex]: [latex]\frac{d}{dx}(x^{1/n}) = \frac{1}{n}x^{(1/n)-1}[/latex]
- For positive integer [latex]n[/latex] and any integer m: [latex]\frac{d}{dx}(x^{m/n}) = \frac{m}{n}x^{(m/n)-1}[/latex]
Use the inverse function theorem to find the derivative of [latex]g(x)=\dfrac{1}{x+2}.[/latex] Compare the result obtained by differentiating [latex]g(x)[/latex] directly.
Use the inverse function theorem to find the derivative of [latex]g(x)=\sqrt[3]{x}[/latex].
Find the derivative of [latex]s(t)=\sqrt{2t+1}[/latex].
Derivatives of Inverse Trigonometric Functions
The Main Idea
- Key Derivatives:
[latex]\begin{array}{lllll}\frac{d}{dx}(\sin^{-1} x)=\large \frac{1}{\sqrt{1-x^2}} & & & & \frac{d}{dx}(\cos^{-1} x)=\large \frac{-1}{\sqrt{1-x^2}} \\ \frac{d}{dx}(\tan^{-1} x)=\large \frac{1}{1+x^2} & & & & \frac{d}{dx}(\cot^{-1} x)=\large \frac{-1}{1+x^2} \\ \frac{d}{dx}(\sec^{-1} x)=\large \frac{1}{|x|\sqrt{x^2-1}} & & & & \frac{d}{dx}(\csc^{-1} x)=\large \frac{-1}{|x|\sqrt{x^2-1}} \end{array}[/latex]
- For composite functions like [latex]\sin^{-1}(g(x))[/latex], use the chain rule in conjunction with inverse trig derivatives
- Pay attention to the domains of inverse trigonometric functions when differentiating
Use the inverse function theorem to find the derivative of [latex]g(x)=\tan^{-1} x[/latex].
Find the derivative of [latex]h(x)=x^2 \sin^{-1} x[/latex]
Find the derivative of [latex]h(x)= \cos^{-1} (3x-1)[/latex]
Find the equation of the line tangent to the graph of [latex]f(x)= \sin^{-1} x[/latex] at [latex]x=0[/latex].