Derivatives as Transformations and Fixed-Point Stability

Grant Sanderson offers a transformational view of derivatives: instead of looking only at the slope of a graph, imagine how a function moves nearby points on a number line. The video is part of Essence of Calculus and includes sponsor support.

Locally, the derivative describes the factor by which a small input displacement is approximately scaled. A negative derivative reverses orientation, and a derivative of zero means the best first-order linear approximation collapses small displacements.

For f(x) = x², the derivative is 2x: around 1 the scale factor is 2, around 3 it is 6, and around −2 it is −4. At zero the local first-order factor is zero.

He applies the idea to repeatedly evaluating f(x) = 1 + 1/x. Solving x = f(x) gives two fixed points: the golden ratio φ and −1/φ. A fixed-point equation alone does not say which limit an iteration will reach.

Starting at 1 gives the familiar positive continued-fraction limit φ. The negative fixed point remains fixed only if started exactly there; nearby values are pushed away. Some starting values, including 0 and −1, make an iterate undefined, so the video’s informal “any starting number” needs that qualification.

A cobweb diagram alternates between the graph of f and the line y = x to show repeated application. Sanderson contrasts it with moving points directly on the number line.

Near φ, the derivative f′(φ) = −1/φ² has magnitude below 1, so nearby displacements shrink. Near −1/φ, the derivative has magnitude above 1, so nearby displacements grow.

For a differentiable one-dimensional iteration, a fixed point with |f′| < 1 is locally attracting, while |f′| > 1 is locally repelling. The boundary case |f′| = 1 needs further analysis. The transformational view extends naturally toward derivatives in higher dimensions.