AP Calculus BC Unit 3 Notes: Derivatives Involving Inverses

Inverse function

A function that reverses the action of another function: if f(x)=y, then f^{-1}(y)=x (when f is one-to-one on its domain).

One-to-one function (injective)

A function where each output corresponds to exactly one input; required so the inverse is a function.

Inverse composition identities

The “undoing” equations for inverses: f(f^{-1}(x))=x and f^{-1}(f(x))=x (on the appropriate domains).

Inverse notation f^{-1}(x)

Means “the inverse function of f evaluated at x,” not a reciprocal and not a power.

Reciprocal of a function value

1/f(x) (or [f(x)]^{-1} as a number); this is NOT the same as the inverse function f^{-1}(x).

Input-output swap property

If f(a)=b, then f^{-1}(b)=a (inverse functions swap inputs and outputs).

Graphical relationship of inverses

Graphs of f and f^{-1} are reflections across the line y=x.

General inverse derivative formula

(f^{-1})'(x)=1 / f'(f^{-1}(x)) (obtained by differentiating f(f^{-1}(x))=x using the chain rule).

Point-specific inverse derivative rule

If f(a)=b and f'(a)≠0, then (f^{-1})'(b)=1/f'(a).

Condition: invertibility for inverse derivatives

To use inverse-derivative ideas, f must be one-to-one on an interval so f^{-1} exists as a function.

Condition: nonzero derivative for inverse slope

If f'(a)=0, then the inverse has an undefined/infinite slope at b=f(a), so a finite (f^{-1})'(b) cannot be found via 1/f'(a).

Cusp/corner issue for inverse differentiability

If f is not differentiable at a (cusp/corner), then f^{-1} may fail to be differentiable at b=f(a) as well.

Common mistake: wrong input for inverse derivative

Using (f^{-1})'(a) instead of (f^{-1})'(b) where b=f(a); the inverse derivative is evaluated at the original output.

Why inverse derivatives are useful (AP context)

They let you find slopes of f^{-1} using values of f and f' from a table/graph without explicitly solving for f^{-1}.

Inverse trigonometric function meaning

An inverse trig function returns an angle whose trig value equals the input (e.g., arcsin(x) is the angle y such that sin(y)=x).

Arcsine (arcsin) principal range

For y=arcsin(x), y is restricted to -π/2 ≤ y ≤ π/2 so the inverse is a function.

Arccosine (arccos) principal range

For y=arccos(x), y is restricted to 0 ≤ y ≤ π so the inverse is a function.

Arctangent (arctan) principal range

For y=arctan(x), y is restricted to -π/2 < y < π/2 so the inverse is a function.

Derivative of arcsin(x)

d/dx[arcsin(x)] = 1/√(1−x^2) (for |x|<1; and defined appropriately at endpoints).

Derivative of arccos(x)

d/dx[arccos(x)] = −1/√(1−x^2).

Derivative of arctan(x)

d/dx[arctan(x)] = 1/(1+x^2).

Derivative of arccot(x)

d/dx[arccot(x)] = −1/(1+x^2) (commonly used in AP Calculus BC).

Derivative of arcsec(x)

d/dx[arcsec(x)] = 1/(|x|√(x^2−1)); the absolute value is essential.

Derivative of arccsc(x)

d/dx[arccsc(x)] = −1/(|x|√(x^2−1)); the absolute value is essential.

Procedure selection principle (derivatives)

Choose the differentiation method based on structure: use inverse-derivative rules for f^{-1}, implicit differentiation when y isn’t isolated (or when you have f(y)=x), and inverse-trig formulas + chain rule when the outer function is arcsin/arccos/arctan/etc.