Understanding Derivatives of Inverses (Including Inverse Trig)

Inverse function

A function that undoes another function: if y=f(x), then x=f^{-1}(y), so f(f^{-1}(x))=x and f^{-1}(f(x))=x.

Defining property of inverses

The pair of identities f(f^{-1}(x))=x and f^{-1}(f(x))=x (on appropriate domains), which characterizes inverse functions.

Reciprocal function (common confusion)

The function 1/f(x); it is NOT the same as f^{-1}(x), which denotes an inverse function.

Reflection across y=x (inverse graphs)

The graph of y=f^{-1}(x) is the reflection of the graph of y=f(x) across the line y=x, swapping x- and y-coordinates.

One-to-one function

A function where each output corresponds to exactly one input (passes the horizontal line test); required for an inverse to be a function.

Differentiability condition for inverse-derivative rule

To use (f^{-1})'(a)=1/f'(f^{-1}(a)), f must be one-to-one and differentiable at the matching point with f'(that point)≠0.

Inverse-derivative formula

If f is differentiable and one-to-one and f'(f^{-1}(x))≠0, then d/dx[f^{-1}(x)]=1/(f'(f^{-1}(x))).

Point-specific inverse derivative

(f^{-1})'(a)=1/f'(f^{-1}(a)); find b such that f(b)=a, then compute 1/f'(b).

Matching points for a function and its inverse

If (b,a) lies on y=f(x) (so f(b)=a), then (a,b) lies on y=f^{-1}(x); these are the corresponding points where slopes reciprocate.

Reciprocal slope idea

The slope of the inverse at x=a equals the reciprocal of the slope of the original at x=b where f(b)=a (inputs swap when inverting).

Vertical tangent (inverse context)

If f'(b)=0, then the inverse has an undefined slope at x=f(b) (a vertical tangent), so (f^{-1})'(f(b)) is not finite/undefined.

Implicit differentiation (inverse functions)

A method to differentiate an inverse relationship by writing f(g(x))=x and differentiating both sides using the chain rule.

Inverse trigonometric function

A function (like arcsin, arccos, arctan) that reverses a trig function after restricting the trig function to a one-to-one principal interval.

Principal range

The restricted output interval used to make an inverse trig function a true function; it determines correct signs (e.g., sign of cos or sin) in derivations.

arcsin(x)

Inverse sine: y=arcsin(x) means sin(y)=x with y in the principal range [-π/2, π/2].

arccos(x)

Inverse cosine: y=arccos(x) means cos(y)=x with y in the principal range [0, π].

arctan(x)

Inverse tangent: y=arctan(x) means tan(y)=x with y in the principal range (-π/2, π/2).

Domain and range of arcsin(x)

Domain [-1,1]; range [-π/2, π/2].

Domain and range of arccos(x)

Domain [-1,1]; range [0, π].

Domain and range of arctan(x)

Domain (-∞,∞); range (-π/2, π/2).

Inverse trig notation warning

sin^{-1}(x), cos^{-1}(x), tan^{-1}(x) mean arcsin(x), arccos(x), arctan(x)—not 1/sin(x), 1/cos(x), 1/tan(x).

Derivative of arcsin(x)

d/dx[arcsin(x)]=1/√(1−x^2) (for |x|<1).

Derivative of arccos(x)

d/dx[arccos(x)]=−1/√(1−x^2) (for |x|<1).

Derivative of arctan(x)

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

Chain rule with inverse trig

If u=u(x), then d/dx[arcsin(u)]=u'/√(1−u^2), d/dx[arccos(u)]=−u'/√(1−u^2), and d/dx[arctan(u)]=u'/(1+u^2).