Sum Rule (Derivative)
The derivative of a sum is the sum of the derivatives.
Formula
Variables
Example
d/dx (x² + x) = 2x + 1.
How it works
If two functions are added together, their rate of change at each point is just the sum of their individual rates of change. This comes directly from the limit definition of the derivative: the difference quotient splits into two separate pieces, one for each function, and limits distribute over addition. So differentiating term-by-term is valid because small changes in each function add up independently.
When to use it
Use this when you have a function written as a sum, such as a polynomial, a combination of trig and algebraic terms, or any expression with separate pieces added together. It is especially useful for simplifying derivatives before applying other rules like the product, quotient, or chain rule.
Worked example, step by step
Find the derivative of f(x) = x^3 + 4x^2 - 7x + 9. Step 1: Differentiate each term separately. d/dx(x^3) = 3x^2 d/dx(4x^2) = 8x d/dx(-7x) = -7 d/dx(9) = 0 Step 2: Add the results. f'(x) = 3x^2 + 8x - 7
Common mistakes to avoid
- forgetting to differentiate every term in the sum
- adding the original functions again instead of their derivatives
- thinking the derivative of a constant is the constant itself
Did You Know?
Derivatives distribute over addition, so you can differentiate a sum one term at a time.
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