Constant Factor Rule (Analysis)

analysis

Definition

Constant Factor Rule

Let $I\subseteq \mathbb{R}$ be an open interval, let $c\in \mathbb{R}$ be a constant, and let $f:I\to \mathbb{R}$ be a differentiable function. Then the scaled function $h(x)=c\cdot f(x)$ is differentiable on $I$ and its derivative is:

$$h^{\prime }(x)=c\cdot f^{\prime }(x)$$

In Leibniz notation:

$$\frac{d}{dx}[c\cdot f(x)]=c\cdot \frac{df}{dx}$$

Intuition

Vertical scaling

Multiplying a function by $c$ stretches or compresses its graph vertically by that factor. Every rise and fall is scaled by $c$, so the slope of the tangent at every point is also scaled by $c$. A steeper graph means proportionally steeper tangents.