A function that 'reverses' another function. If $f(a) = b$, then $f^{-1}(b) = a$.

A function is differentiable at a point if its derivative exists at that point.

A function that has an inverse function.

The derivative of a function $f(x)$ is a measure of how $f(x)$ changes as $x$ changes.

A line that touches a curve at a point and has the same slope as the curve at that point.

The equation of a line given a point $(x_1, y_1)$ and slope $m$: $y - y_1 = m(x - x_1)$.

The set of all possible input values (x-values) for which the function is defined.

The set of all possible output values (y-values) of the function.

A function $f(x)$ is strictly increasing if, for any $x_1 < x_2$, we have $f(x_1) < f(x_2)$.

The reciprocal of a number $x$ is $1/x$.

The derivative of the inverse function at a point is the reciprocal of the derivative of the original function at the corresponding point. If $f(a) = b$, then $(f^{-1})'(b) = \frac{1}{f'(a)}$.

The graphs of a function and its inverse are reflections of each other across the line $y = x$.

The derivative of a function at a point represents the slope of the tangent line to the function's graph at that point.

A strictly increasing or decreasing function is guaranteed to be one-to-one, and therefore invertible.

It represents the derivative of the original function evaluated at the inverse function, ensuring the correct corresponding point is used for the reciprocal calculation.

At a sufficiently small scale, a differentiable function can be approximated by its tangent line.

The domain of $f(x)$ is the range of $f^{-1}(x)$, and the range of $f(x)$ is the domain of $f^{-1}(x)$.

The original function must be differentiable at the point corresponding to the inverse function's input for the inverse derivative to exist.

$f(x)$ is the original function, and $f^{-1}(x)$ is its inverse, which 'undoes' the operation of $f(x)$.

$f'(x)$ is the derivative of the original function, and $(f^{-1})'(x)$ is the derivative of its inverse.

If $f(x)$ is increasing, $f'(x) > 0$, so $(f^{-1})'(x) > 0$ as well, meaning the graph of $(f^{-1})'(x)$ is positive.

The graph of the inverse function is the reflection of the original function across the line $y = x$.

A vertical tangent line on $f(x)$ means $f'(x) = 0$ at that point, which implies the derivative of the inverse function is undefined (has a vertical asymptote) at the corresponding point.

The concavity of $f(x)$ affects the rate of change of $f'(x)$, which in turn affects the shape of $(f^{-1})'(x)$. A concave up $f(x)$ may lead to a different shape for $(f^{-1})'(x)$ compared to a concave down $f(x)$.

It means $f(x)$ is an odd function, i.e., $f(-x) = -f(x)$.

If $f'(x) > 0$, then $f(x)$ is increasing, and $f^{-1}(x)$ is also increasing. If $f'(x) < 0$, then $f(x)$ is decreasing, and $f^{-1}(x)$ is also decreasing.

A sharp corner in $f(x)$ means it's not differentiable at that point, which can affect the differentiability of $f^{-1}(x)$ at the corresponding point.

The domain of $f^{-1}(x)$ is the range of $f(x)$, and the range of $f^{-1}(x)$ is the domain of $f(x)$.

A horizontal asymptote in $f(x)$ becomes a vertical asymptote in $f^{-1}(x)$.

If $f(x)$ is linear, $f'(x)$ is constant, so $(f^{-1})'(x)$ is also constant.