Skill: Differentiate sums and differences

Explanation and Free Practice Resources

Differentiation is linear: for differentiable functions, the derivative of a sum is the sum of the derivatives, and the derivative of a difference is the difference of the derivatives, $(f+g)'=f'+g'$ and $(f-g)'=f'-g'$ . This allows a function’s rate of change or tangent slope to be found term by term, while preserving signs and recognizing that constant terms contribute zero; more advanced operator or abstract treatments are outside this scope.

On this page:

Learn “Differentiate sums and differences” with a detailed explanation of the concept
Master “Differentiate sums and differences” with unlimited, free practice problems
Download (or modify) a free worksheet on Differentiate sums and differences you can use at home or in class with practice problems that scaffold and explore “Differentiate sums and differences” from different perspectives.
Start working on adaptive units like “Basic Derivatives - Practice” or “Derivative Rules - Intro” that introduce, teach, or practice “Differentiate sums and differences” to maximize your learning.

Detailed Explanation: Differentiate sums and differences

To differentiate a sum or difference, differentiate each term separately and keep the signs unchanged:

(f+g)'=f'+g' \qquad\text{and}\qquad (f-g)'=f'-g'.

Remember:

Use the power rule: $\dfrac{d}{dx}(x^n)=nx^{n-1}$ .
The derivative of a constant is $0$ .

Example: Find the derivative of

y=3x^4-5x^2+7x-9.

Differentiate each term one at a time:

\frac{d}{dx}(3x^4)=12x^3,

\frac{d}{dx}(-5x^2)=-10x,

\frac{d}{dx}(7x)=7,

and

\frac{d}{dx}(-9)=0.

Now combine the results, preserving the original signs:

\boxed{y'=12x^3-10x+7}.

The constant term $-9$ disappears because constants do not change as $x$ changes.

Learn by doing: Differentiate sums and differences

Click a topic below to practice the foundational skills you'll need, learn the steps, or master this skill

Your results

Practice with unlimited practice problems

Derivative Rules - Sum Rule Positive Powers to Derivative

Earned ?

Download/Modify Free Grade 12 'Differentiate sums and differences' Worksheet

Print this free 'Differentiate sums and differences' worksheet to practice this skill away from the computer. Click 'Modify' to easily customize this worksheet to your exact needs.

Units Teaching This Skill

These adaptive units introduce, teach, or reinforce this skill (Differentiate sums and differences) through scaffolded problems that grow in difficulty as concepts are understood.