📝 Derivative of constant function (11 MCQs)

📖 From Calculus • 3. The Derivation • 11 questions available

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What is Derivative of constant function?

Definition:
The derivative of a constant function is always zero because a constant does not change with respect to the variable, meaning its rate of change is nonexistent, resulting in a horizontal line with zero slope across the entire domain.

Example:
For $f(x) = 7$ , the derivative is $f'(x) = 0$ since there is no variation in output regardless of input changes.

Reason:
This basic rule simplifies differentiation of polynomial terms and serves as a foundation for understanding that only varying components contribute to rates of change in composite functions.

Easy

Medium

Hard

📝 All Derivative of constant function MCQs

Q1. What is $\frac{d}{dx}[7]$ ?

A.0 ✅

B.7

C.1

D.undefined

💡 Difficulty: easy | ✅ Correct: A

📖 Explanation: The derivative of any constant is zero because the definition $\frac{d}{dx}[c]=\lim_{h\to0}\frac{c-c}{h}$ gives $\lim_{h\to0}0=0$ . Thus $\frac{d}{dx}[7]=0$ , reflecting that a horizontal line has no steepness.

Q2. If $f(x)=c$ is a constant, which of the following must be true about the slope of its tangent line at any point?

A.It is positive

B.It is zero ✅

C.It is negative

D.IT DEPENDS ON

X

💡 Difficulty: medium | ✅ Correct: B

Q3. Suppose $g(x)=c$ and $h(x)=3g(x)+5$ . What is $h'(x)$ ?

A.0 ✅

B.3

C.5

D.15

💡 Difficulty: medium | ✅ Correct: A

📖 Explanation: First compute $h(x)=3c+5$ , which is still a constant because $c$ does not depend on $x$ . The derivative of any constant is zero, so $h'(x)=0$ . Options involving 3,5, or 15 would arise only if the expression contained $x$ .

Q4. Which statement correctly compares the derivative of $f(x)=c$ and the derivative of $p(x)=mx+b$ (with $m\neq0$ )?

A.Both are zero

B.The derivative of

f

is zero while that of

p

equals

m

✅

C.Both equal

m

D.The derivative of

f

equals

m

while that of

p

is zero

💡 Difficulty: medium | ✅ Correct: B

📖 Explanation: For a constant function $f(x)=c$ , the derivative is always zero. For a linear function $p(x)=mx+b$ , the derivative is the constant coefficient $m$ . Hence the correct comparison is that $f'$ is zero and $p'$ equals $m$ .

Q5. Given the function $F(x)=(x^{2})\cdot c$ where $c$ is a constant, what is $F'(x)$ ?

2xc

✅

2c

C.0

X^{2}

💡 Difficulty: hard | ✅ Correct: A

Q6. If a constant $k$ is added to a differentiable function $f(x)$ , how does the derivative change?

A.It increases by

k

B.It decreases by

k

C.It remains unchanged ✅

D.IT BECOMES

K

💡 Difficulty: medium | ✅ Correct: C

Q7. Which principle explains why the derivative of any constant function is zero?

A.Mean Value Theorem

B.Fundamental Theorem of Calculus

C.Definition of derivative as limit of difference quotient ✅

D.Chain Rule

💡 Difficulty: easy | ✅ Correct: C

📖 Explanation: The definition of the derivative as $\lim_{h\to0}\frac{f(x+h)-f(x)}{h}$ directly yields zero when $f(x)=c$ because the numerator is always $c-c=0$ . This limit‑based reasoning is the fundamental principle behind the result.

Q8. Using the derivative of a constant, evaluate $\frac{d}{dx}\bigl[\sin(x)+\pi\bigr]$ ?

\cos(x)

✅

-\cos(x)

\sin(x)

D.0

💡 Difficulty: medium | ✅ Correct: A

Q9. Let $f(x)=c\cdot e^{x}$ . Applying the product rule, what is $f'(x)$ ?

c\cdot e^{x}

✅

e^{x}

C.0

D.C

💡 Difficulty: hard | ✅ Correct: A

Q10. If the derivative of a function at every point is zero, which statement is necessarily true?

A.The function is constant everywhere ✅

B.The function has a horizontal asymptote

C.The function is linear

D.THE FUNCTION IS PERIODIC

💡 Difficulty: hard | ✅ Correct: A

Q11. If you differentiate the constant term in the polynomial $4x^{3}-2x+7$ , what contribution does it make to the derivative?

A.7

B.0 ✅

C.-2

D.4

💡 Difficulty: easy | ✅ Correct: B

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