Show that the slopes of the graphs of f and f^-1 are...

Show that the slopes of the graphs of $f$ and $f^{-1}$ are reciprocals at the given points.
$$\begin{aligned}
&f(x)=x^{3} \quad \quad \left(\frac{1}{2}, \frac{1}{8}\right) \\
&f^{-1}(x)=\sqrt[3]{x} \quad \quad \left(\frac{1}{8}, \frac{1}{2}\right)
\end{aligned}$$

Key Concepts

Chain Rule

The chain rule is a fundamental tool in calculus used to differentiate composite functions. In the context of inverse functions, it facilitates understanding of how the derivative of the composition of a function and its inverse yields a value of one, thereby underpinning the reciprocal relationship between the derivatives of the function and its inverse.

Inverse Function Derivative Formula

This formula states that the derivative of the inverse function at a point is the reciprocal of the derivative of the original function at the corresponding point. Mathematically, if f is differentiable and has a differentiable inverse f?¹, then (f?¹)'(y) = 1/(f'(x)) where x and y are related by y = f(x). This relation is crucial when analyzing how the slopes of a function and its inverse are interconnected.

Inverse Function

An inverse function essentially reverses the effect of the original function, mapping outputs back to their respective inputs. In other words, if f maps an input x to an output y, then its inverse f?¹ maps y back to x. This concept is foundational in understanding how functions can undo each other’s operations.

Derivative

The derivative of a function at a given point represents the slope of the tangent line to the graph of the function at that point. This slope is a measure of how the function’s output changes in response to a small change in its input, providing insight into the function’s instantaneous rate of change.

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