📝 Derivative definition first principle calculus (20 MCQs)

📖 From Calculus • 3. The Derivation • 20 questions available

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What is Derivative definition first principle calculus?

Definition:
The derivative defined by first principles uses the limit definition $f'(x) = \lim_{h \to 0} \frac{f(x+h) - f(x)}{h}$ to calculate the instantaneous rate of change, establishing the rigorous mathematical foundation for differential calculus operations.

Example:
For $f(x) = x^2$ , $f'(x) = \lim_{h \to 0} \frac{(x+h)^2 - x^2}{h} = \lim_{h \to 0} (2x+h) = 2x$ .

Reason:
This method ensures conceptual clarity by deriving rules from basics rather than memorization, helping students understand why derivative formulas work and building strong analytical reasoning skills.

Easy

Medium

Hard

📝 All Derivative definition first principle calculus MCQs

Q1. According to the definition, how is the derivative $f'(x)$ expressed as a limit?

f&#039;(x)=\displaystyle\lim_{h\to 0}\frac{f(x+h)-f(x)}{h}

✅

f&#039;(x)=\displaystyle\lim_{h\to 0}\frac{f(x)-f(x-h)}{h}

f&#039;(x)=\displaystyle\lim_{h\to 0}\frac{f(x)-f(x+h)}{h}

f&#039;(x)=\displaystyle\lim_{h\to 0}\frac{f(x+h)+f(x)}{h}

💡 Difficulty: easy | ✅ Correct: A

📖 Explanation: The definition of the derivative states that the instantaneous rate of change at a point is the limit of the difference quotient as $h$ approaches zero. Option A matches this definition exactly, while the other options alter the numerator or sign, resulting in incorrect expressions. Hence, A is the correct choice.

Q2. What is the name given to the expression $\frac{f(x+h)-f(x)}{h}$ in the definition of the derivative?

A.Difference quotient ✅

B.Derivative limit

C.Instantaneous slope

D.Tangent ratio

💡 Difficulty: easy | ✅ Correct: A

📖 Explanation: The term $\frac{f(x+h)-f(x)}{h}$ is traditionally called the difference quotient because it measures the average rate of change over an interval of width $h$ . It is the central building block for forming the derivative via a limit. The other names are not standard terminology for this expression.

Q3. If a function $f$ is differentiable at $x=1$ and $f'(1)=0$ , which of the following must be true about the graph of $f$ near $x=1$ ?

A.The tangent line is horizontal at

x=1

. ✅

B.The function attains a maximum at

x=1

C.The function is constant on an interval containing

x=1

D.The graph must be symmetric about

x=1

💡 Difficulty: easy | ✅ Correct: A

📖 Explanation: A zero derivative indicates that the slope of the tangent line at that point is zero, meaning the tangent line is horizontal. This does not guarantee a maximum, constancy, or symmetry; those require additional information about the second derivative or the function’s behavior. Therefore, only statement A is guaranteed.

Q4. Suppose $g(x)=\sqrt{x}$ . Using the definition of derivative, which of the following statements correctly explains why $g'(4)=\frac{1}{4}$ ?

A.The limit

\lim_{h\to0}\frac{\sqrt{4+h}-2}{h}

simplifies to

\frac{1}{4}

B.The limit

\lim_{h\to0}\frac{4+h-4}{h}

gives

\frac{1}{4}

. ✅

C.The derivative of

\sqrt{x}

is always

\frac{1}{2\sqrt{x}}

, evaluated at

x=4

D.The function

\sqrt{x}

is linear near

x=4

💡 Difficulty: medium | ✅ Correct: B

📖 Explanation: Evaluating the limit directly shows $\frac{\sqrt{4+h}-2}{h}=\frac{h}{(\sqrt{4+h}+2)h}$ which approaches $\frac{1}{4}$ as $h\to0$ . Option B misapplies the definition, C uses a known formula without justification, and D incorrectly claims linearity. Hence, A provides the correct logical reasoning.

Q5. If a function $f$ is differentiable on an interval and its derivative $f'(x)$ is always positive, what can be inferred about the original function $f$ on that interval?

f

is strictly increasing. ✅

f

has a global maximum.

f

is constant.

f

is concave upward.

💡 Difficulty: medium | ✅ Correct: A

📖 Explanation: A positive derivative at every point means the slope of the tangent line never falls below zero, implying the function rises as $x$ increases. This guarantees strict monotonic increase. It does not imply a maximum, constancy, or concavity, which depend on the sign of the second derivative or other conditions. Therefore, statement A is correct.

Q6. Consider two functions $f$ and $g$ with $f'(x)=g'(x)$ for all $x$ in their common domain. Which conclusion is logically valid?

f(x)=g(x)+C

for some constant

C

. ✅

f

and

g

are identical functions.

f

and

g

have the same curvature.

f

and

g

intersect at exactly one point.

💡 Difficulty: medium | ✅ Correct: A

📖 Explanation: If the derivatives of two functions are identical everywhere, their difference $h(x)=f(x)-g(x)$ has derivative zero, meaning $h$ is constant. Hence $f(x)=g(x)+C$ . Equality of functions, curvature, or intersection count cannot be deduced without additional information. Option A captures the valid inference.

Q7. A function $p$ satisfies $p'(x)=2p(x)$ for every $x$ . Which of the following must be true about the shape of its graph?

A.Its graph is an exponential curve. ✅

B.Its graph is a straight line through the origin.

C.Its graph is a parabola.

D.Its graph is periodic.

💡 Difficulty: hard | ✅ Correct: A

📖 Explanation: The differential equation $p'(x)=2p(x)$ solves to $p(x)=Ce^{2x}$ , an exponential function whose rate of change is proportional to its current value. This produces a characteristic exponential growth shape, not a line, parabola, or periodic pattern. Therefore, the graph must be exponential.

Q8. Given $f(x)=x^3-3x$ , which statement correctly uses the definition of derivative to determine where the tangent line is horizontal?

A.Solve

\displaystyle\lim_{h\to0}\frac{(x+h)^3-3(x+h)-\bigl(x^3-3x\bigr)}{h}=0

. ✅

B.Set

3x^2-3=0

and solve for

x

C.Find

x

where

f(x)=0

D.Find

x

where

f&#039;&#039;(x)=0

💡 Difficulty: medium | ✅ Correct: A

📖 Explanation: The definition requires evaluating the limit of the difference quotient and setting it to zero to locate points with zero slope. While the simplified derivative $3x^2-3$ yields the same result, the question explicitly asks for a method using the limit definition, making option A the correct logical approach.

Q9. If $f$ is differentiable at $x=0$ and satisfies $\displaystyle\lim_{h\to0}\frac{f(h)-f(0)}{h}=5$ , which of the following is necessarily true?

f&#039;(0)=5

. ✅

f

is linear near

0

f(0)=5

f

has a maximum at

0

💡 Difficulty: medium | ✅ Correct: A

📖 Explanation: The limit given is precisely the definition of the derivative at $x=0$ . Hence $f'(0)=5$ . The other statements require extra conditions such as linearity, specific function values, or extremum behavior, none of which follow from the limit alone.

Q10. For a function defined by $f(x)=\begin{cases}x^2,&x\le 1\\2x-1,&x>1\end{cases}$ , which of the following correctly determines whether $f$ is differentiable at $x=1$ ?

A.Check that the left-hand and right-hand limits of the difference quotient both exist and are equal. ✅

B.Compute

f&#039;(1)

using the power rule.

C.Observe that the function is continuous at

x=1

D.Differentiate each piece and compare the formulas at

x=1

💡 Difficulty: hard | ✅ Correct: A

📖 Explanation: Differentiability at a point where a piecewise definition meets requires the limit of the difference quotient from both sides to exist and match. Continuity alone is insufficient; the derivative formulas must also agree. Option A describes the precise logical test, whereas the other options miss the necessary two-sided limit condition.

Q11. Suppose $h(x)=\ln(x^2+1)$ . Using the definition of derivative, which reasoning correctly leads to the derivative $h'(x)=\frac{2x}{x^2+1}$ ?

A.Rewrite the difference quotient, rationalize the numerator, and take the limit as

h\to0

. ✅

B.Apply the chain rule directly without justification.

C.Differentiate

\ln(u)

\frac{1}{u}

and then multiply by the derivative of

u=x^2+1

D.Assume the derivative of

\ln x

\frac{1}{x}

and substitute

x^2+1

for

x

💡 Difficulty: medium | ✅ Correct: A

📖 Explanation: The definition approach requires forming $\frac{\ln((x+h)^2+1)-\ln(x^2+1)}{h}$ , using logarithmic properties to combine terms, rationalizing, and then evaluating the limit, which yields $\frac{2x}{x^2+1}$ . Options B, C, and D invoke rules without the limit justification required by the question.

Q12. Which of the following best describes why the derivative function $f'(x)$ can be viewed as a “slope-producing” function?

A.For each

x

f&#039;(x)

gives the slope of the tangent line to the graph of

f

at that point. ✅

B.It tells how fast

f

grows as

x

increases.

C.It provides the area under the curve of

f

D.It indicates the curvature of the graph of

f

💡 Difficulty: easy | ✅ Correct: A

📖 Explanation: The term “slope-producing” emphasizes that the value of $f'(x)$ at a particular $x$ is precisely the slope of the tangent line to the curve $y=f(x)$ at that abscissa. While the derivative also reflects growth rate, the specific geometric interpretation highlighted by the phrase is the tangent slope, making option A correct.

Q13. A student claims that because the limit $\displaystyle\lim_{h\to0}\frac{f(x+h)-f(x)}{h}$ exists, the function $f$ must be continuous at $x$ . Which analysis correctly evaluates this claim?

A.The claim is true; differentiability implies continuity. ✅

B.The claim is false; a function can have a limit of the difference quotient without being continuous.

C.Continuity is unrelated to the existence of the limit.

D.The claim holds only for polynomial functions.

💡 Difficulty: medium | ✅ Correct: A

📖 Explanation: If the limit defining the derivative exists, the function must be continuous at that point because the difference quotient forces $f(x+h)$ to approach $f(x)$ as $h\to0$ . Hence differentiability guarantees continuity, validating the student's claim. The other options misstate the relationship.

Q14. Consider the function $k(x)=|x|$ . Which of the following correctly uses the definition of derivative to explain why $k$ is not differentiable at $x=0$ ?

A.The left-hand limit of the difference quotient equals

-1

while the right-hand limit equals

1

. ✅

B.The derivative of

|x|

\frac{x}{|x|}

everywhere.

C.The function is not continuous at

0

D.The graph has a cusp at

0

💡 Difficulty: hard | ✅ Correct: A

📖 Explanation: Applying the definition, the limit from the left yields $\lim_{h\to0^-}\frac{|h|-0}{h}=-1$ and from the right yields $\lim_{h\to0^+}\frac{|h|-0}{h}=1$ . Since the two one-sided limits differ, the overall limit does not exist, so $k$ is not differentiable at $0$ . The other statements either misapply the derivative or address unrelated properties.

Q15. If a function $m$ satisfies $m'(x)=m(x)$ for all real $x$ and $m(0)=3$ , which conceptual step allows you to determine $m(x)$ without computing limits?

A.Recognize that the differential equation solves to

m(x)=3e^{x}

. ✅

B.Apply the power rule to infer

m(x)=3x

C.Assume

m

is linear because its derivative equals itself.

D.Use the definition of the derivative to write a limit expression for

m(x)

💡 Difficulty: medium | ✅ Correct: A

📖 Explanation: The condition $m'(x)=m(x)$ is a first‑order linear differential equation whose unique solution, given the initial value $m(0)=3$ , is the exponential function $m(x)=3e^{x}$ . This conceptual insight bypasses direct limit computation. The other options misuse calculus rules or ignore the exponential nature.

Q16. A function $p$ is defined by $p(x)=\frac{1}{x}$ for $x\neq0$ . Using the definition of derivative, which reasoning correctly shows that $p$ is not differentiable at $x=0$ ?

A.The difference quotient

\frac{1/(x+h)-1/x}{h}

does not have a limit as

h\to0

because the function is undefined at

x=0

. ✅

B.Since

p

is not defined at

0

, it cannot be differentiable there.

C.The derivative would be

-1/x^{2}

, which is undefined at

0

D.All of the above.

💡 Difficulty: hard | ✅ Correct: A

📖 Explanation: The definition requires evaluating $\lim_{h\to0}\frac{p(0+h)-p(0)}{h}$ , but $p(0)$ does not exist, making the difference quotient undefined and preventing any limit from forming. While the other statements are true, option A directly addresses the failure of the limit definition, providing the correct logical explanation.

Q17. Which of the following statements best synthesizes the relationship between the derivative function $f'(x)$ and the original function $f(x)$ in terms of limiting processes?

f&#039;(x)

is obtained by taking the limit of the average rate of change of

f

over an interval that shrinks to a point. ✅

f&#039;(x)

is the integral of

f

over an infinitesimal interval.

f&#039;(x)

measures the total change of

f

from a fixed point to

x

f&#039;(x)

is the second derivative of

f

💡 Difficulty: medium | ✅ Correct: A

📖 Explanation: The derivative emerges from the limit of the difference quotient, which represents the average rate of change of $f$ over a small interval. As the interval length $h$ approaches zero, this average becomes the instantaneous rate of change, captured by $f'(x)$ . The other options mischaracterize the derivative’s definition.

Q18. Given that $f'(x)=2x$ for all real $x$ , which conceptual inference about the original function $f$ is valid?

f(x)=x^{2}+C

for some constant

C

. ✅

f(x)=2x+C

f(x)

must be a linear function.

f(x)

is undefined at

x=0

💡 Difficulty: medium | ✅ Correct: A

📖 Explanation: Integrating the derivative $2x$ yields $f(x)=x^{2}+C$ , where $C$ is an arbitrary constant reflecting the family of antiderivatives. This aligns with the fundamental relationship between a function and its derivative. The other options either misinterpret the integration or introduce incorrect properties.

Q19. If a function $s$ satisfies $s'(x)=\cos(x)$ and $s(0)=5$ , which logical deduction correctly determines $s\left(\frac{\pi}{2}\right)$ ?

s\left(\frac{\pi}{2}\right)=5+\int_{0}^{\pi/2}\cos(t)\,dt=5+1

. ✅

s\left(\frac{\pi}{2}\right)=5+\sin\left(\frac{\pi}{2}\right)

s\left(\frac{\pi}{2}\right)=5+\cos\left(\frac{\pi}{2}\right)

s\left(\frac{\pi}{2}\right)=5

💡 Difficulty: hard | ✅ Correct: A

📖 Explanation: Since $s'(x)=\cos(x)$ , integrating from $0$ to $\pi/2$ gives the change in $s$ : $\int_{0}^{\pi/2}\cos(t)dt = \sin(\pi/2)-\sin(0)=1$ . Adding this to the initial value $5$ yields $s(\pi/2)=6$ . Option A correctly applies the Fundamental Theorem of Calculus, while the others misuse the antiderivative or neglect the integral.

Q20. A function $r$ is defined by $r(x)=x^4$ . Which of the following correctly uses the definition of derivative to show that the tangent line at $x=1$ has slope $4$ ?

A.Evaluate

\displaystyle\lim_{h\to0}\frac{(1+h)^4-1}{h}=4

. ✅

B.Apply the power rule to obtain

r&#039;(x)=4x^3

and substitute

x=1

C.Differentiate

x^4

using the product rule.

D.Assume the slope is the coefficient of the highest-degree term.

💡 Difficulty: easy | ✅ Correct: A

📖 Explanation: Using the limit definition, $\frac{(1+h)^4-1}{h} = \frac{4h+6h^2+4h^3+h^4}{h}=4+6h+4h^2+h^3$ . As $h\to0$ , the expression approaches $4$ , confirming the tangent slope at $x=1$ . The other options rely on rules rather than the explicit limit process requested.

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