Sum Rule (Analysis)

analysis

Definition

Sum Rule

Let $I\subseteq \mathbb{R}$ be an open interval and let $f,g:I\to \mathbb{R}$ be differentiable functions. Then the sum $h(x)=f(x)+g(x)$ is differentiable on $I$ and its derivative is:

$$h^{\prime }(x)=f^{\prime }(x)+g^{\prime }(x)$$

In Leibniz notation:

$$\frac{d}{dx}[f(x)+g(x)]=\frac{df}{dx}+\frac{dg}{dx}$$

Intuition

Adding slopes

The derivative measures the rate of change of a function. If two functions change simultaneously, the total change is simply the sum of the individual changes. A small step $\Delta x$ produces changes $\Delta f\approx f^{\prime }(x)\Delta x$ and $\Delta g\approx g^{\prime }(x)\Delta x$, so:

$$\Delta (f+g)\approx (f^{\prime }(x)+g^{\prime }(x))\Delta x$$

Dividing by $\Delta x$ and taking the limit gives the sum rule.