Inverse Function Derivatives

Unit 3 · Differentiation — Composite, Implicit, Inverse

Differentiating Inverse Functions

The derivative of an inverse function is the reciprocal of the derivative of the original function at the corresponding point. For inverse trig, the co- functions (arccos, arccot, arccsc) are the negatives of the non-co versions.

Inverse Function Derivative

If (a, b) is on f, then (b, a) is on f⁻¹ and

\left(f^{-1}\right)'(b) = \frac{1}{f'(a)} = \frac{1}{f'\!\left(f^{-1}(b)\right)}

Inverse Trig Derivatives (AP Required)

\frac{d}{dx}[\arcsin x] = \frac{1}{\sqrt{1-x^2}}

\frac{d}{dx}[\arccos x] = \frac{-1}{\sqrt{1-x^2}}

\frac{d}{dx}[\arctan x] = \frac{1}{1+x^2}

With Chain Rule

\frac{d}{dx}[\arctan(u)] = \frac{u'}{1+u^2} \qquad \frac{d}{dx}[\arcsin(u)] = \frac{u'}{\sqrt{1-u^2}}

AP Tip: For (f⁻¹)′(b): first find a = f⁻¹(b) from the table (which x gives f(x) = b?), then compute 1/f′(a).

Type 1

Inverse Function Derivative from a Table

Use the table to identify (f⁻¹)(b) = a, then compute 1/f′(a).

Example 1

The table gives values of f. Find (f⁻¹)′(7).

Example 2

f(2) = 5, f′(2) = 1/3. Find (f⁻¹)′(5).

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Type 2

arctan and arcsin with Chain Rule

Identify u and u′, then apply the inverse trig derivative formula.

Example 3

Find the derivative.

\frac{d}{dx}[\arctan(3x^2)]

Example 4

Find the derivative.

\frac{d}{dx}\!\left[\arcsin\!\left(\frac{x}{2}\right)\right]

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Type 3

Inverse Trig + Product Rule

When an inverse trig function appears in a product, combine the product rule with the arctan/arcsin derivative.

Example 5

Find f′(x).

f(x) = x^2 \arctan x

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