📝 Power rule for derivatives (18 MCQs)

📖 From Calculus • 3. The Derivation • 18 questions available

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What is Power rule for derivatives?

Definition:
The power rule states that for any real number $n$ , the derivative of $x^n$ is $nx^{n-1}$ , providing a straightforward method to differentiate polynomial terms by multiplying by the exponent and reducing the power by one.

Example:
For $f(x) = x^5$ , applying the power rule gives $f'(x) = 5x^{5-1} = 5x^4$ directly without using limit definitions.

Reason:
This rule dramatically speeds up differentiation of polynomials, making it efficient to handle higher-degree terms and forming the basis for differentiating more complex algebraic expressions in calculus.

Easy

Medium

Hard

📝 All Power rule for derivatives MCQs

Q1. What is the derivative of $x^3$ ?

3x^2

✅

2x

x^3

3x

💡 Difficulty: easy | ✅ Correct: A

📖 Explanation: Applying the power rule $d/dx[x^n]=nx^{n-1}$ with $n=3$ yields $3x^{2}$ . This matches option A, confirming the rule works for integer exponents and giving the correct derivative of the cubic function.

Q2. According to the extended power rule, the derivative of $x^{1/2}$ is?

2x^{1/2}

-\frac12 x^{-3/2}

\frac12 x^{1/2}

\frac12 x^{-1/2}

✅

💡 Difficulty: easy | ✅ Correct: D

📖 Explanation: The extended power rule states $d/dx[x^{r}]=r x^{r-1}$ . For $r=\tfrac12$ we have $\tfrac12 x^{-1/2}$ , which is option D. This shows the rule also applies to fractional exponents, producing a reciprocal‑square‑root factor.

Q3. If $f(x)=x^{n}$ with $n>0$ , what can be inferred about the sign of $f'(x)$ for $x>0$ ?

A.Negative

B.Positive ✅

C.Zero

D.Depends on

n

💡 Difficulty: medium | ✅ Correct: B

📖 Explanation: Using the power rule, $f'(x)=n x^{n-1}$ . Since both $n$ and $x^{n-1}$ are positive for $x>0$ , the product is positive, so the derivative is positive for all such $x$ . This logical inference follows directly from the rule.

Q4. For $g(x)=x^{-4}$ , what is the behavior of its derivative as $x\to\infty$ ?

A.Approaches 0 ✅

B.Approaches

-\infty

C.Approaches

\infty

D.Oscillates

💡 Difficulty: medium | ✅ Correct: A

📖 Explanation: The derivative is $g'(x)=-4x^{-5}$ . As $x$ grows large, the term $x^{-5}$ tends to zero, making the whole expression approach zero. Hence the derivative diminishes to 0, illustrating how negative powers yield vanishing slopes at infinity.

Q5. If $h(x)=x^{\pi}$ , how does the growth of $h(x)$ compare to its derivative as $x\to\infty$ ?

A.Derivative grows faster

B.Same order ✅

C.Function grows faster

D.Both are constant

💡 Difficulty: hard | ✅ Correct: B

📖 Explanation: Both $h(x)=x^{\pi}$ and $h'(x)=\pi x^{\pi-1}$ are power functions differing only by a constant factor $\pi$ and an exponent reduced by one. As $x$ becomes large, the dominant term is the power of $x$ ; thus they increase at the same polynomial order.

Q6. Compare the derivatives of $x^{4}$ and $x^{5}$ at $x=2$ .

A.Derivative of

x^{5}

larger ✅

B.Derivative of

x^{4}

larger

C.Equal

D.Depends on sign

💡 Difficulty: medium | ✅ Correct: A

📖 Explanation: Using the power rule: $(x^{4})' =4x^{3}=4\cdot8=32$ and $(x^{5})' =5x^{4}=5\cdot16=80$ . At $x=2$ , the derivative of the quintic is larger, confirming option A. This analytical comparison shows how higher exponents produce steeper slopes.

Q7. Which power function has the smallest magnitude of derivative at $x=1$ : $x^{-2},\,x^{1/3},\,x^{2},\,x^{0}$ ?

x^{-2}

x^{1/3}

x^{2}

x^{0}

✅

💡 Difficulty: easy | ✅ Correct: D

📖 Explanation: The derivative of a constant $x^{0}$ is zero, giving the smallest possible magnitude. All other choices have non‑zero derivatives at $x=1$ . Hence option D correctly identifies the constant function as having the minimal derivative magnitude.

Q8. For $p(x)=x^{7/2}$ , what is the second derivative?

\frac{35}{4}x^{3/2}

✅

\frac{35}{4}x^{5/2}

\frac{21}{4}x^{3/2}

\frac{21}{4}x^{5/2}

💡 Difficulty: hard | ✅ Correct: A

📖 Explanation: First derivative: $(7/2)x^{5/2}$ . Differentiating again gives $(7/2)(5/2)x^{3/2}=35/4\,x^{3/2}$ . This matches option A, demonstrating repeated application of the power rule to obtain higher‑order derivatives.

Q9. Given $f(x)=x^{n}$ with integer $n$ , which statement about $f'(x)$ is always true?

f&#039;(x)

is a polynomial of degree

n-1

✅

f&#039;(x)

is always positive

f&#039;(x)

has the same zeros as

f(x)

f&#039;(x)

is constant

💡 Difficulty: medium | ✅ Correct: A

📖 Explanation: Applying the power rule yields $f'(x)=n x^{\,n-1}$ , a polynomial whose highest exponent is $n-1$ . This property holds for any integer $n$ , confirming option A as the universally correct statement.

Q10. Why does the power rule hold for non‑integer exponents using the limit definition?

A.Binomial expansion works for rational exponents

B.Continuity of the exponent function

C.Both A and B ✅

D.None of the above

💡 Difficulty: hard | ✅ Correct: C

📖 Explanation: The limit definition expands $(x+h)^{r}$ via the generalized binomial series, which is valid for any real $r$ . Combined with the continuity of the function $x^{r}$ , the higher‑order terms vanish, leaving $r x^{r-1}$ . Thus both reasons are needed, making option C correct.

Q11. If $f(x)=x^{r}$ with $r<0$ , what does the derivative indicate about behavior near $x=0$ ?

A.Derivative tends to

\infty

in magnitude ✅

B.Derivative tends to 0

C.Derivative is undefined

D.Derivative is constant

💡 Difficulty: medium | ✅ Correct: A

📖 Explanation: The derivative is $f'(x)=r x^{r-1}$ . With $r<0$ , the exponent $r-1$ is even more negative, so as $x\to0^{+}$ , $x^{r-1}$ blows up, and the product with the finite constant $r$ diverges to $\pm\infty$ . Hence the magnitude grows without bound.

Q12. How would you differentiate $f(x)=\dfrac{(3x^{2}+2)^{5}}{x^{3}}$ using the power rule?

A.Apply the rule to numerator and denominator separately

B.Rewrite as a product of powers and apply ✅

C.Use the quotient rule only

D.Cannot apply the power rule

💡 Difficulty: hard | ✅ Correct: B

📖 Explanation: Rewrite the function as $(3x^{2}+2)^{5}\,x^{-3}$ . Then apply the chain rule to the first factor and the power rule to the second, treating the whole expression as a product. This approach leverages the power rule within a more complex context, matching option B.

Q13. Find the derivative of $g(t)=t^{\sqrt{2}}$ at $t=1$ .

\sqrt{2}

✅

1

\sqrt{2}\,t^{\sqrt{2}-1}

0

💡 Difficulty: easy | ✅ Correct: A

📖 Explanation: Using the power rule, $g'(t)=\sqrt{2}\,t^{\sqrt{2}-1}$ . Evaluating at $t=1$ gives $\sqrt{2}\cdot1^{\sqrt{2}-1}=\sqrt{2}$ . Thus the derivative at $t=1$ equals $\sqrt{2}$ , which is option A.

Q14. Compare the derivatives of $x^{1/3}$ and $\sqrt{x}$ for $x>0$ .

A.First derivative larger for all

x

B.Second derivative larger for all

x

C.They are equal at

x=1

D.They are equal at

x=(2/3)^{6}

✅

💡 Difficulty: medium | ✅ Correct: D

📖 Explanation: Set $(1/3)x^{-2/3}=(1/2)x^{-1/2}$ and solve for $x$ . This yields $x=(2/3)^{6}\approx0.088$ . Hence the two derivatives intersect only at this specific value, making option D the correct statement.

Q15. If $h(x)=x^{2}\cdot x^{3/4}$ , what is its derivative?

\frac{11}{4}x^{7/4}

✅

\frac{11}{4}x^{5/4}

\frac{11}{4}x^{11/4}

\frac{11}{4}x^{1/4}

💡 Difficulty: easy | ✅ Correct: A

📖 Explanation: Combine the powers: $x^{2}x^{3/4}=x^{11/4}$ . Differentiating gives $(11/4)x^{7/4}$ , which corresponds to option A. This demonstrates how the power rule simplifies products of like bases.

Q16. Suppose a function satisfies $f'(x)=5x^{4}$ . Which of the following could be $f(x)$ ?

x^{5}

x^{5}+7

\frac{5}{5}x^{5}

x^{5}+C

✅

💡 Difficulty: easy | ✅ Correct: D

📖 Explanation: Integrating $5x^{4}$ yields $x^{5}+C$ , where $C$ is an arbitrary constant. Options A, B, and C are special cases, but the most general form is option D, reflecting the constant of integration.

Q17. Find the derivative of $(2x)^{3}$ .

6(2x)^{2}

12x^{2}

24x^{2}

✅

6x^{2}

💡 Difficulty: easy | ✅ Correct: C

📖 Explanation: First rewrite $(2x)^{3}=8x^{3}$ . Differentiating gives $24x^{2}$ . This matches option C, showing that expanding before differentiating yields the same result as applying the chain rule directly.

Q18. Determine the integer $n$ such that the derivative of $x^{n}$ equals $6x^{5}$ .

n=6

✅

n=5

n=7

n=6

(duplicate)

💡 Difficulty: hard | ✅ Correct: A

📖 Explanation: Applying the power rule: $(x^{n})'=n x^{n-1}=6x^{5}$ . Equating exponents gives $n-1=5\Rightarrow n=6$ , and the coefficient also matches. Hence $n=6$ is the unique solution, corresponding to option A.

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