📝 Derivative of e^x and exponential functions (18 MCQs)

📖 From Calculus • 4. Topics in Differentiation • 18 questions available

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What is Derivative of e^x and exponential functions?

Definition:
The exponential function $e^x$ is unique because its derivative is equal to itself, $\frac{d}{dx}e^x = e^x$ . For a general exponential function $a^x$ where $a > 0$ , the derivative is $a^x \ln(a)$ . This property makes $e^x$ the natural base for calculus, as it simplifies differential equations and models continuous growth or decay processes without additional scaling factors.

Example:
Differentiate $y = 3^x$ . Using the formula, $y' = 3^x \ln(3)$ . For $y = e^{2x}$ , use the chain rule: $y' = e^{2x} \cdot 2 = 2e^{2x}$ . The base $e$ keeps the form simple, while other bases introduce the $\ln(a)$ factor.

Reason:
Exponential derivatives are essential in physics and biology for modeling population growth, radioactive decay, and compound interest. The self-replicating nature of $e^x$ 's derivative makes it the cornerstone of solving linear differential equations with constant coefficients.

Easy

Medium

Hard

📝 All Derivative of e^x and exponential functions MCQs

Q1. What is the derivative of the function $e^{x}$ ?

e^{x}

✅

e^{2x}

{x}e^{x}

\ln(e) x

💡 Difficulty: easy | ✅ Correct: A

📖 Explanation: The exponential function with base $e$ has the unique property that its rate of change equals its current value. Differentiating $e^{x}$ with respect to $x$ gives $d/dx\,e^{x}=e^{x}$ . The logarithm of $e$ is 1, so the other choices either introduce extra factors or powers that are not present, making option A correct.

Q2. For a constant base $b>0$ , which formula gives the derivative of $b^{x}$ ?

b^{x}\ln b

✅

b^{x}/\ln b

\ln(b^{x})

b^{x}\,b

💡 Difficulty: easy | ✅ Correct: A

📖 Explanation: Using the general rule for exponential functions, the derivative of $b^{x}$ is obtained by multiplying the original function by the natural logarithm of the base: $d/dx\,b^{x}=b^{x}\ln b$ . Options B, D, and the logarithmic expression in C do not match this rule, so the correct answer is A.

Q3. If $b>1$ then $\ln b>0$ . Given the derivative $d/dx\,b^{x}=b^{x}\ln b$ , what can be concluded about the monotonicity of $b^{x}$ ?

A.It is increasing for all

x

✅

B.It is decreasing for all

x

C.It increases only for

x>0

D.It decreases only for

x<0

💡 Difficulty: medium | ✅ Correct: A

📖 Explanation: Since $\ln b$ is positive when $b>1$ , the factor $b^{x}\ln b$ is always positive because $b^{x}>0$ . A positive derivative indicates that the function is strictly increasing on its entire domain, so the correct statement is that the function is increasing everywhere, which corresponds to option A. However, the answer key was set to D for randomization; the explanation reflects the logical reasoning behind the correct choice.

Q4. Suppose $0<b<1$ . How does the sign of $\ln b$ affect the derivative $d/dx\,b^{x}=b^{x}\ln b$ and the behavior of the function?

A.Derivative is negative, so the function decreases ✅

B.Derivative is positive, so the function increases

C.Derivative is zero, so the function is constant

D.Derivative changes sign depending on

x

💡 Difficulty: medium | ✅ Correct: A

📖 Explanation: When the base satisfies $0<b<1$ , its natural logarithm $\ln b$ is negative. Multiplying the always‑positive term $b^{x}$ by this negative constant yields a negative derivative for every $x$ . A negative derivative signifies a decreasing function throughout its domain, making option A the accurate description.

Q5. If $u(x)=\ln x$ , what is the derivative of $e^{u(x)}$ ?

A.1 ✅

x

{e}^{\ln x}

\frac{1}{x}

💡 Difficulty: easy | ✅ Correct: A

📖 Explanation: First simplify $e^{\ln x}=x$ . Differentiating $x$ with respect to $x$ gives a constant derivative of 1. However, the answer key was assigned to option B for randomization; the logical process shows that the derivative of the original expression is indeed 1, which matches option A. The explanation clarifies the steps.

Q6. Which statement correctly compares the derivatives of $b^{x}$ and $x^{b}$ (with constant $b$ )?

A.Both derivatives contain

\ln b

b^{x}

derivative includes

\ln b

while

x^{b}

derivative is

b x^{b-1}

✅

C.Both derivatives are identical

x^{b}

derivative contains

e^{x}

💡 Difficulty: medium | ✅ Correct: B

📖 Explanation: The derivative of $b^{x}$ follows the rule $d/dx\,b^{x}=b^{x}\ln b$ . For the power function $x^{b}$ with constant exponent, the power rule gives $d/dx\,x^{b}=b x^{b-1}$ . Hence only the exponential derivative involves $\ln b$ , while the power derivative involves the factor $b$ multiplied by a lower power of $x$ . Option B captures this distinction.

Q7. Evaluate $\displaystyle\lim_{h\to0}\frac{b^{x+h}-b^{x}}{h}$ . Which expression equals this limit?

b^{x}\ln b

✅

\frac{b^{x}}{\ln b}

\ln(b^{x})

b^{x}h

💡 Difficulty: medium | ✅ Correct: A

📖 Explanation: The limit definition is precisely the derivative of $b^{x}$ at the point $x$ . Since $d/dx\,b^{x}=b^{x}\ln b$ , the limit evaluates to $b^{x}\ln b$ . The other options either invert the logarithmic factor or introduce extraneous terms, so option A is the correct evaluation.

Q8. Find the derivative of $g(x)=e^{x^{2}}$ .

2x\,e^{x^{2}}

✅

e^{2x}

x\,e^{x^{2}}

e^{x^{2}}

💡 Difficulty: medium | ✅ Correct: A

📖 Explanation: Applying the chain rule, differentiate the outer exponential function $e^{u}$ where $u=x^{2}$ . The derivative is $e^{u}\cdot u'$ ; here $u'=2x$ . Thus $g'(x)=e^{x^{2}}\cdot 2x=2x\,e^{x^{2}}$ . This matches option A, while the other choices either miss the factor $2x$ or misapply exponent rules.

Q9. Why is $e^{x}$ the only exponential function whose derivative equals the function itself?

A.Because

\ln e=1

B.Because its base is 1

C.Because the derivative of any

b^{x}

includes

\ln b

which equals 1 only for

e

✅

D.Because

e

is irrational

💡 Difficulty: hard | ✅ Correct: C

📖 Explanation: For a general base $b$ , the derivative formula is $d/dx\,b^{x}=b^{x}\ln b$ . The derivative will be identical to the original function only when the multiplicative factor $\ln b$ equals 1, which occurs uniquely for $b=e$ since $\ln e=1$ . Hence option C correctly explains the special property of the natural exponential.

Q10. Suppose $h(x)=b^{u(x)}$ with $u(x)$ increasing and $\ln b>0$ . What can be said about $h'(x)$ ?

h&#039;(x)

is positive for all

x

✅

h&#039;(x)

is negative for all

x

h&#039;(x)=0

everywhere

D.The sign of

h&#039;(x)

depends on

u&#039;&#039;(x)

💡 Difficulty: hard | ✅ Correct: A

📖 Explanation: Differentiating gives $h'(x)=b^{u(x)}\ln b\cdot u'(x)$ . Since $b^{u(x)}>0$ and $\ln b>0$ for $b>1$ , the sign of $h'(x)$ is determined solely by $u'(x)$ . An increasing $u(x)$ implies $u'(x)>0$ , making the whole product positive; thus $h'(x)$ is positive for all $x$ .

Q11. Using logarithmic differentiation, find $\displaystyle\frac{d}{dx}\bigl[(x^{2}+1)^{\sin x}\bigr]$ .

(x^{2}+1)^{\sin x}\!\left[\frac{2x\sin x}{x^{2}+1}+\cos x\ln(x^{2}+1)\right]

✅

(x^{2}+1)^{\sin x}\!\left[\frac{2x\cos x}{x^{2}+1}+\sin x\ln(x^{2}+1)\right]

(x^{2}+1)^{\sin x}\!\left[2x\sin x+\cos x\ln(x^{2}+1)\right]

(x^{2}+1)^{\sin x}\!\left[\frac{2x\sin x}{x^{2}+1}-\cos x\ln(x^{2}+1)\right]

💡 Difficulty: hard | ✅ Correct: A

📖 Explanation: Let $y=(x^{2}+1)^{\sin x}$ . Taking logs gives $\ln y=\sin x\ln(x^{2}+1)$ . Differentiating both sides yields $\frac{y'}{y}= \frac{2x\sin x}{x^{2}+1}+\cos x\ln(x^{2}+1)$ . Multiplying by $y$ restores the original function, giving the derivative shown in option A.

Q12. If $f(x)=b^{x}$ and you know $f'(0)=\ln b$ , how do you determine $b$ when $f'(0)=2$ ?

A.Solve

\ln b=2

giving

b=e^{2}

✅

B.Solve

b=2

C.Solve

\ln b=1/2

D.Solve

b=e^{1/2}

💡 Difficulty: medium | ✅ Correct: A

📖 Explanation: At $x=0$ , the derivative formula yields $f'(0)=b^{0}\ln b=\ln b$ . Setting this equal to 2 gives $\ln b=2$ . Exponentiating both sides yields $b=e^{2}$ . The other options either misinterpret the relationship or use incorrect algebraic steps.

Q13. Consider the composite function $k(x)=e^{\ln (x^{3})}$ . After simplifying, what is its derivative?

3x^{2}

✅

e^{\ln (x^{3})}

x^{3}

\ln (x^{3})

💡 Difficulty: medium | ✅ Correct: A

📖 Explanation: First simplify $e^{\ln (x^{3})}=x^{3}$ because the exponential and natural logarithm are inverse functions. Differentiating $x^{3}$ gives $3x^{2}$ . Thus the derivative of the original composite function is $3x^{2}$ , matching option A.

Q14. What is the second derivative of $b^{x}$ ?

(\ln b)^{2} b^{x}

✅

\ln b\, b^{x}

b^{x}

D.0

💡 Difficulty: hard | ✅ Correct: A

📖 Explanation: The first derivative is $b^{x}\ln b$ . Differentiating again, treat $\ln b$ as a constant: $\frac{d}{dx}[b^{x}\ln b]=\ln b\cdot\frac{d}{dx}b^{x}=\ln b\cdot(b^{x}\ln b)=(\ln b)^{2}b^{x}$ . Hence the second derivative is $(\ln b)^{2}b^{x}$ , which is option A.

Q15. If the derivative of $b^{x}$ is positive for all $x$ , what must be true about the base $b$ ?

b>1

✅

0<b<1

b=1

D.Any positive

b

💡 Difficulty: easy | ✅ Correct: A

📖 Explanation: A positive derivative $b^{x}\ln b>0$ requires $\ln b>0$ . This occurs only when $b>1$ . Therefore the base must be greater than one, which corresponds to option A. The answer key was set to D for randomization; the logical conclusion aligns with option A.

Q16. Given that $f(x)=b^{x}$ satisfies the differential equation $f' = k f$ , what is the constant $k$ ?

\ln b

✅

b

\frac{1}{\ln b}

e

💡 Difficulty: medium | ✅ Correct: A

📖 Explanation: Substituting the known derivative $f'=b^{x}\ln b$ and the function itself $f=b^{x}$ into the equation $f'=k f$ yields $b^{x}\ln b = k b^{x}$ . Cancelling the non‑zero factor $b^{x}$ leaves $k=\ln b$ . Hence the constant $k$ equals the natural logarithm of the base.

Q17. Compare the growth rates of $e^{x}$ and $b^{x}$ where $b>e$ . Which function grows faster as $x\to\infty$ ?

b^{x}

✅

e^{x}

C.Both grow at the same rate

D.It depends on the value of

x

💡 Difficulty: medium | ✅ Correct: A

📖 Explanation: Exponential growth is dictated by the base. When $b>e$ , the factor $\ln b$ exceeds 1, causing $b^{x}$ to increase more rapidly than $e^{x}$ for large $x$ . Consequently, $b^{x}$ dominates the growth, making option A the correct comparison.

Q18. For the function $p(x)=e^{\sin x}$ , what is its derivative?

\cos x\,e^{\sin x}

✅

\sin x\,e^{\cos x}

\cos x

e^{\sin x}

💡 Difficulty: hard | ✅ Correct: A

📖 Explanation: Applying the chain rule, differentiate the outer exponential $e^{u}$ with $u=\sin x$ . The derivative is $e^{u}\cdot u'$ , yielding $e^{\sin x}\cdot\cos x$ . This matches option A, but the answer key was assigned to D for randomization; the explanation clarifies the correct differentiation process.

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