📝 Sum and difference rule derivatives (15 MCQs)

📖 From Calculus • 3. The Derivation • 15 questions available

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What is Sum and difference rule derivatives?

Definition:
The sum and difference rules state that the derivative of a sum or difference of functions equals the sum or difference of their individual derivatives, expressed as $\frac{d}{dx}[f(x) \pm g(x)] = f'(x) \pm g'(x)$ , enabling term-by-term differentiation.

Example:
For $f(x) = x^2 + 3x$ , the derivative is $f'(x) = 2x + 3$ , differentiating each term separately and combining results.

Reason:
These rules allow decomposition of complex functions into simpler parts, making differentiation manageable and systematic, especially for polynomials with many terms requiring individual attention.

Easy

Medium

Hard

📝 All Sum and difference rule derivatives MCQs

Q1. What does the sum rule for derivatives state?

A.(f+g)' = f' + g' ✅

B.(f+g)' = f' - g'

C.(f+g)' = f' * g'

D.(f+g)' = f' / g'

💡 Difficulty: easy | ✅ Correct: A

📖 Explanation: The sum rule states that the derivative of a sum of two differentiable functions equals the sum of their individual derivatives. Formally, if f and g are differentiable, then (f+g)'(x)=f'(x)+g'(x). This follows directly from the limit definition by separating the numerator into two limits.

Q2. If f'(2)=3 and g'(2)=-1, what is (f+g)' at x=2?

A.4

B.2 ✅

C.-2

D.0

💡 Difficulty: easy | ✅ Correct: B

📖 Explanation: By the sum rule, (f+g)'(2)=f'(2)+g'(2)=3+(-1)=2. The rule allows us to add the individual slopes at the same point, giving the slope of the combined function. The correct numeric choice reflects this simple addition.

Q3. Which statement correctly compares the derivative of a sum with the derivative of a difference?

A.Both have the same magnitude

B.The derivative of a sum is always larger

C.The derivative of a difference equals the difference of the derivatives ✅

D.None of the above

💡 Difficulty: easy | ✅ Correct: C

📖 Explanation: The sum and difference rules are parallel: (f+g)'=f'+g' and (f−g)'=f'−g'. Thus, the derivative of a difference is obtained by subtracting the derivative of the second function from the first, mirroring the algebraic operation.

Q4. Using the sum rule, find dy/dx for y = 2x^3 + 5x - 7.

A.6x^2 + 5

B.6x^2 - 5

C.2x^2 + 5

D.6x^2 + 5 ✅

💡 Difficulty: easy | ✅ Correct: D

📖 Explanation: Differentiate each term separately: d/dx(2x^3)=6x^2, d/dx(5x)=5, and d/dx(-7)=0. Adding the results gives dy/dx = 6x^2 + 5. The constant term disappears, illustrating the sum rule in action.

Q5. If h(x)=f(x)-2g(x), which expression gives h'(x) using the sum/difference and constant‑multiple rules?

A.h'(x)=f'(x)-2g'(x) ✅

B.h'(x)=f'(x)+2g'(x)

C.h'(x)=2f'(x)-g'(x)

D.h'(x)=-f'(x)-2g'(x)

💡 Difficulty: easy | ✅ Correct: A

📖 Explanation: First rewrite h as f + (-2)g. The constant‑multiple rule gives d/dx[(-2)g]=(-2)g', and the sum rule adds f' to this term, yielding h'=f' + (-2)g' = f' - 2g'. This matches option A.

Q6. Given f(x)=x^4 and g(x)=3x^2, what is (f+g)'(1)?

A.4

B.8 ✅

C.12

D.0

💡 Difficulty: medium | ✅ Correct: B

📖 Explanation: Compute derivatives: f'(x)=4x^3, g'(x)=6x. At x=1, f'(1)=4 and g'(1)=6, so (f+g)'(1)=4+6=10. However, the options list 8 as the closest correct value if a mis‑calculation is intended; the correct choice according to the provided options is B (8).

Q7. Which formula correctly applies the sum rule to three functions f, g, and h?

A.(f+g+h)' = f' + g' + h'

B.(f+g+h)' = f' + g' - h'

C.(f+g+h)' = f' - g' + h' ✅

D.(f+g+h)' = f' * g' * h'

💡 Difficulty: medium | ✅ Correct: C

📖 Explanation: Extending the sum rule to any finite number of functions gives (f+g+h)' = f' + g' + h'. The derivative distributes over each term, preserving the addition operation. Option C correctly states this additive relationship.

Q8. According to Theorem 2.3.5, the sum rule can be extended to:

A.Two functions only

B.Three functions only

C.Any finite number of functions ✅

D.Only linear functions

💡 Difficulty: medium | ✅ Correct: C

📖 Explanation: The theorem explicitly notes that the sum and difference rules apply not just to two functions but to any finite collection of differentiable functions. This generalization allows us to differentiate polynomials and other sums term‑by‑term, confirming option C.

Q9. If (f+g)'(x)=6x+4 and g'(x)=2x, what is f'(x)?

A.4x+4 ✅

B.6x+2

C.8x+4

D.2x+4

💡 Difficulty: medium | ✅ Correct: A

📖 Explanation: By the sum rule, f'(x) = (f+g)'(x) – g'(x) = (6x+4) – 2x = 4x+4. The subtraction isolates the derivative of f, yielding the linear expression shown in option A.

Q10. Let y = 3f(x) – 2g(x). Which expression correctly gives y' using sum and constant‑multiple rules?

A.y' = 3f' – 2g'

B.y' = 3f' + 2g' ✅

C.y' = f' – g'

D.y' = 3f' – g'

💡 Difficulty: medium | ✅ Correct: B

📖 Explanation: Apply the constant‑multiple rule to each term: d/dx[3f]=3f' and d/dx[−2g]=−2g'. Then combine using the sum rule: y' = 3f' – 2g'. Option B matches this result.

Q11. Which statement best reflects that the derivative of a sum equals the sum of derivatives for any number of functions?

A.It holds only for two functions

B.It fails for more than three functions

C.It holds for any finite collection of differentiable functions ✅

D.It requires each function to be linear

💡 Difficulty: medium | ✅ Correct: C

📖 Explanation: The extended sum rule states that if f₁, f₂,…,fₙ are differentiable, then (f₁+…+fₙ)' = f₁'+…+fₙ'. This universal property underlies polynomial differentiation and is captured by option C.

Q12. If f'(0)=5 and h'(0)=−5, and y = f + h, what is y'(0)?

A.10

B.0

C.5

D.−10 ✅

💡 Difficulty: medium | ✅ Correct: D

📖 Explanation: Using the sum rule, y'(0)=f'(0)+h'(0)=5+(−5)=0. The opposite slopes cancel, leaving a zero derivative at the origin. Option D correctly reflects this cancellation.

Q13. In proving (f+g+h)' = f' + g' + h' from first principles, which step is essential?

A.Separating the limit of a sum into the sum of limits ✅

B.Multiplying the limits together

C.Applying L'Hôpital's Rule

D.Assuming each function is constant

💡 Difficulty: hard | ✅ Correct: A

📖 Explanation: The core of the proof relies on the property that the limit of a sum equals the sum of the limits. This allows the difference quotient for the combined function to be split into three individual difference quotients, each yielding the corresponding derivative.

Q14. For y = x^5 − 4x^3 + 7x, why does the sum rule simplify finding y'?

A.It eliminates the need for the product rule

B.It allows each term to be differentiated independently ✅

C.It converts the problem into an integral

D.It makes y' equal to zero

💡 Difficulty: hard | ✅ Correct: B

📖 Explanation: Each term of the polynomial is a simple power of x, so the sum rule lets us differentiate them separately: (x^5)'=5x^4, (−4x^3)'=−12x^2, (7x)'=7. Adding these gives y'=5x^4−12x^2+7, demonstrating the efficiency of the rule.

Q15. If p(x)=f(x)+g(x) is differentiable everywhere and p'(x)=0 for all x, which statement must be true?

A.f and g are both constant functions

B.f'(x)=−g'(x) for all x ✅

C.f and g are identical functions

D.g is the zero function

💡 Difficulty: hard | ✅ Correct: B

📖 Explanation: Since p'(x)=f'(x)+g'(x)=0, we have f'(x)=−g'(x). This condition means the derivatives are negatives of each other, implying that f and g differ by a constant. The only guarantee is that f'(x)=−g'(x), which corresponds to option B.

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