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Sum, Difference, and Constant Multiple Rules

Combine the power rule with simple rules for sums, differences, and constants multiplying a function.

Combining Terms

Extra differentiation rules
Extra differentiation rules

Together with the power rule, these three rules let you differentiate almost any expression made of added or subtracted power terms — one term at a time.

Example 1 — a sum of two terms

Differentiate f(x) = 3 x^{2} + 5 x . 1 Differentiate each term separately. 2 Derivative of 3 x^{2} : keep the 3, apply the power rule to x^{2} \to 3 \times 2 x = 6 x 3 Derivative of 5 x : keep the 5, apply the power rule to x \to 5 \times 1 = 5 4 Add the results: 6 x + 5 Answer: f'(x) = 6 x + 5

Example 2 — a difference of two terms

Differentiate f(x) = 4 x^{3} - 2 x . 1 Derivative of 4 x^{3} : 4 \times 3 x^{2} = 12 x^{2} 2 Derivative of 2 x : 2 \times 1 = 2 3 Subtract: 12 x^{2} - 2 Answer: f'(x) = 12 x^{2} - 2

Practice

What is the derivative of 5 x^{2} - 3 x + 1 ?

Combining Terms

Extra differentiation rules
Extra differentiation rules

Together with the power rule, these three rules let you differentiate almost any expression made of added or subtracted power terms — one term at a time.

Example 1 — a sum of two terms

Differentiate f(x) = 3 x^{2} + 5 x . 1 Differentiate each term separately. 2 Derivative of 3 x^{2} : keep the 3, apply the power rule to x^{2} \to 3 \times 2 x = 6 x 3 Derivative of 5 x : keep the 5, apply the power rule to x \to 5 \times 1 = 5 4 Add the results: 6 x + 5 Answer: f'(x) = 6 x + 5

Example 2 — a difference of two terms

Differentiate f(x) = 4 x^{3} - 2 x . 1 Derivative of 4 x^{3} : 4 \times 3 x^{2} = 12 x^{2} 2 Derivative of 2 x : 2 \times 1 = 2 3 Subtract: 12 x^{2} - 2 Answer: f'(x) = 12 x^{2} - 2

Practice

What is the derivative of 5 x^{2} - 3 x + 1 ?