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 $(f^{-1})'(a)=\frac{1}{f'(f^{-1}(a))}$, $f$ must be one-to-one and differentiable at the matching point with $f'(\text{that point})≠0$.

Inverse-derivative formula

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

Point-specific inverse derivative

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

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 $[-\frac{\theta}{2}, \frac{\theta}{2}]$.

arccos(x)

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

arctan(x)

Inverse tangent: $y=\arctan(x)$ means $\tan(y)=x$ with $y$ in the principal range $(-\frac{\pi}{2}, \frac{\pi}{2})$.

Domain and range of arcsin(x)

Domain $[-1,1]$; range $[-\frac{\theta}{2}, \frac{\theta}{2}]$.

Domain and range of arccos(x)

Domain $[-1,1]$; range $[0, \theta]$.

Domain and range of arctan(x)

Domain $(-\infty,\infty)$; range $(-\frac{\pi}{2}, \frac{\pi}{2})$.

Inverse trig notation warning

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

Derivative of arcsin(x)

$\frac{d}{dx}[\arcsin(x)]=\frac{1}{\sqrt{1-x^2}} \text{ (for } |x|<1\text{)}$.

Derivative of arccos(x)

$\frac{d}{dx}[\arccos(x)]=-\frac{1}{\sqrt{1-x^2}}$ (for $|x|<1$).

Derivative of arctan(x)

$\frac{d}{dx}[\arctan(x)]=\frac{1}{1+x^2}$ (for all real $x$).

Chain rule with inverse trig

If $u=u(x)$, then $\frac{d}{dx}[\arcsin(u)]=\frac{u'}{\sqrt{1-u^2}}$, $\frac{d}{dx}[\arccos(u)]=-\frac{u'}{\sqrt{1-u^2}}$, and $\frac{d}{dx}[\arctan(u)]=\frac{u'}{(1+u^2)}$.