Sum and Difference Rule

The sum rule implies that differentiation is a linear operation. This means you can break down a complex, multi-term equation and find the derivative of the entire function by simply differentiating each term individually and adding the results together.

The sum rule states that for any two differentiable functions f(x) and g(x):

$\frac{d}{dx}\left( f\,\,+\,\,g \right) \,\,=\,\,\frac{d}{dx}\left( f \right) \,\,+\,\,\frac{d}{dx}\left( g \right)$

This rule also applies to subtraction (the difference rule):

$\frac{d}{dx}\left( f\,\,-\,\,g \right) \,\,=\,\,\frac{d}{dx}\left( f \right) \,\,-\,\,\frac{d}{dx}\left( g \right)$

References

Calaway, Shana, Dale Hoffman, & David Lippman. “2.4: Power and Sum Rules for Derivatives.” LibreTexts, June 27, 2021. https://math.libretexts.org/Bookshelves/Calculus/Applied_Calculus_(Calaway_Hoffman_and_Lippman)/02%3A_The_Derivative/2.04%3A_Power_and_Sum_Rules_for_Derivatives.

[ ℰ ] “Introduction to Differentiation.” 2.3 Sum rule. Open Learning. Accessed June 30, 2026. https://www.open.edu/openlearn/science-maths-technology/introduction-differentiation/content-section-4.3.

Khandelwal, Neha. “Differentiation Rules: Sums and Differences.” CK-12 Foundation, July 4, 2025. https://flexbooks.ck12.org/cbook/ck-12-cbse-maths-class-11/section/12.8/primary/lesson/differentiation-rules%3A-sums-and-differences/.

“Sum Rule.” Statistics How To. August 18, 2020. https://www.statisticshowto.com/derivatives/sum-rule/.

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Sum & Difference Rule

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