How does the sum rule for derivatives follow from stacking output heights?

Conditions

• Functions $f$ and $g$ are differentiable.
• The input increment is common to both functions.

Reasoning, step by step

1. Represent $f(x)$ and $g(x)$ as stacked vertical segments.
2. Apply a small input change $dx$.
3. Observe that the change in the total height is $\Delta f + \Delta g$.
4. Divide by $dx$ to get the average rate of change: $\frac{\Delta f}{dx} + \frac{\Delta g}{dx}$.
5. Take the limit as $dx \to 0$.
6. Conclude that $(f+g)' = f' + g'$.

Example

For $\sin x + x^2$, the derivative is $\cos x + 2x$. The change in the sine part and the change in the square part add up directly.

Common misconceptions

• Confusing the sum rule with statistical independence of random variables.
• Thinking that the sum rule requires the functions to be positive; it holds for signed values as well.
• Believing that the derivative of a sum is the product of derivatives.

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• What is the derivative of $\sin x + x^2$ according to the sum rule?