Derivatives of Inverse Trigonometric Functions

Learning Outcomes

  • Recognize the derivatives of the standard inverse trigonometric functions

We now turn our attention to finding derivatives of inverse trigonometric functions. These derivatives will prove invaluable in the study of integration later in this text. The derivatives of inverse trigonometric functions are quite surprising in that their derivatives are actually algebraic functions. Previously, derivatives of algebraic functions have proven to be algebraic functions and derivatives of trigonometric functions have been shown to be trigonometric functions. Here, for the first time, we see that the derivative of a function need not be of the same type as the original function.

Example: Derivative of the Inverse Sine Function

Use the inverse function theorem to find the derivative of [latex]g(x)=\sin^{-1} x[/latex].

Example: Applying the Chain Rule to the Inverse Sine Function

Apply the chain rule to the formula derived in Example: Applying the Inverse Function Theorem
to find the derivative of [latex]h(x)=\sin^{-1} (g(x))[/latex] and use this result to find the derivative of [latex]h(x)=\sin^{-1}(2x^3)[/latex].

Watch the following video to see the worked solution to Example: Applying the Chain Rule to the Inverse Sine Function.

Try It

Use the inverse function theorem to find the derivative of [latex]g(x)=\tan^{-1} x[/latex].

The derivatives of the remaining inverse trigonometric functions may also be found by using the inverse function theorem. These formulas are provided in the following theorem.

Derivatives of Inverse Trigonometric Functions


[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]

 

Example: Applying Differentiation Formulas to an Inverse Tangent Function

Find the derivative of [latex]f(x)=\tan^{-1} (x^2)[/latex]

Example: Applying Differentiation Formulas to an Inverse Sine Function

Find the derivative of [latex]h(x)=x^2 \sin^{-1} x[/latex]

Watch the following video to see the worked solution to Example: Applying Differentiation Formulas to an Inverse Sine Function.

Try It

Find the derivative of [latex]h(x)= \cos^{-1} (3x-1)[/latex]

Example: Applying the Inverse Tangent Function

The position of a particle at time [latex]t[/latex] is given by [latex]s(t)= \tan^{-1}\left(\dfrac{1}{t}\right)[/latex] for [latex]t\ge \frac{1}{2}[/latex]. Find the velocity of the particle at time [latex]t=1[/latex].

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Find the equation of the line tangent to the graph of [latex]f(x)= \sin^{-1} x[/latex] at [latex]x=0[/latex].

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