Unit 2.7 Practice Activity MCQs: Exponential, Logarithmic, and Trigonometric Derivatives
A full revision page with original study notes, exam-style multiple-choice practice, and concise worked solutions for exponential, logarithmic, and trigonometric derivative questions.
What You Need to Know for Unit 2.7
This Unit 2.7 practice page focuses on a common cluster of AP Calculus skills: limits involving exponential functions, derivative definitions connected to \(a^x\) and \(\ln x\), and direct differentiation of expressions involving \(\sin x\), \(\cos x\), and \(e^x\). Many students lose marks here not because the rules are difficult, but because the questions are written in disguised forms.
For example, a limit may really be asking for the derivative of an exponential function at a point, or a logarithmic expression may be written in a way that introduces a negative sign. This page is designed to make those patterns easy to recognize before you even start calculating.
Best revision method: read the study notes first, attempt the A form of a question without opening the solution, then solve the B form to confirm that you truly understand the method rather than just memorizing the answer.
Study Notes: Exponential, Logarithmic, and Trigonometric Derivative Patterns
1) Standard exponential limit
A key fact is
\[
\lim_{x\to 0}\frac{a^x-1}{x}=\ln a \quad \text{for } a>0.
\]
This is one of the fastest ways to identify limits involving exponential functions. If the expression looks slightly different, try rewriting it so this standard form appears.
2) Derivative of an exponential function
If \(f(x)=a^x\), then
\[
f'(x)=a^x\ln a.
\]
So a limit like
\[
\lim_{h\to 0}\frac{a^{c+h}-a^c}{h}
\]
is often just \(f'(c)\) for \(f(x)=a^x\).
3) Negative signs in logarithmic difference quotients
If you see
\[
\lim_{x\to a}\frac{f(a)-f(x)}{x-a},
\]
be careful. This equals \(-f'(a)\), not \(f'(a)\). For \(f(x)=\ln x\), the derivative is \(f'(x)=\frac{1}{x}\), so the sign matters immediately.
4) Trig and exponential derivatives at \(x=0\)
The values \(\sin 0=0\), \(\cos 0=1\), and \(e^0=1\) make many derivatives at zero very efficient. Always simplify using exact trig values before checking the answer choices.
Core idea: before doing algebra, classify the question. Is it a standard limit, a derivative at a point, a disguised difference quotient, or a direct derivative evaluation?
Exam Strategy and Common Mistakes
These questions are usually short, but they punish rushed reading. Use this quick checklist while solving.
Look for the form \( \frac{a^x-1}{x} \) or a version that can be rewritten into it.
If a limit uses \(h \to 0\), check whether it is the derivative of a function at a fixed input.
For logarithmic difference quotients, track the sign carefully.
For trig and exponential derivatives, use exact values such as \(\sin 0\), \(\cos 0\), and \(e^0\).
If the limit is \(0/0\), think about whether differentiation or a known identity is the quickest path.
Common errors: forgetting the factor caused by a transformed input such as \(2x\), losing a negative sign in a logarithmic quotient, and choosing \(\ln a\) when the full expression should really be multiplied by another constant.
Practice Activity MCQs
Work without calculators unless your teacher allows one. These questions are designed to test recognition of derivative structure, not just symbolic differentiation. Click “Show solution” only after making a genuine attempt.
Rewrite the expression as
\[
\frac{4^{2x}-1}{x}=2\cdot \frac{4^{2x}-1}{2x}.
\]
Then apply the standard limit:
\[
\lim_{x\to 0}\frac{4^{2x}-1}{2x}=\ln 4.
\]
Therefore the full limit is
\[
2\ln 4.
\]
Recognize the derivative at \(x=1\) of \(f(x)=5^x\):
\[
\lim_{h\to 0}\frac{5^{1+h}-5^1}{h}=f'(1).
\]
Since
\[
f'(x)=5^x\ln 5,
\]
we get
\[
f'(1)=5\ln 5.
\]
This is the derivative of \(7^x\) evaluated at \(x=2\):
\[
\lim_{h\to 0}\frac{7^{2+h}-7^2}{h}= \left.\frac{d}{dx}(7^x)\right|_{x=2}.
\]
Since
\[
\frac{d}{dx}(7^x)=7^x\ln 7,
\]
the value is
\[
7^2\ln 7=49\ln 7.
\]
Q 3-A.
Let \(f(x)=\ln x\). Evaluate \( \displaystyle \lim_{x\to 1}\frac{f(1)-f(x)}{x-1} \).
A. \(-1\)
B. \(-\tfrac{1}{2}\)
C. \(0\)
D. \(1\)
Answer: A
For a differentiable function,
\[
\lim_{x\to a}\frac{f(a)-f(x)}{x-a}=-f'(a).
\]
Here \(f(x)=\ln x\), so
\[
f'(x)=\frac{1}{x}.
\]
Hence
\[
-f'(1)=-1.
\]
Q 3-B.
Let \(f(x)=\ln x\). Evaluate \( \displaystyle \lim_{x\to e}\frac{\ln(e)-\ln(x)}{x-e} \).
A. \(-\dfrac{1}{e}\)
B. \(\dfrac{1}{e}\)
C. \(-1\)
D. \(1\)
Answer: A
Apply the same identity:
\[
\lim_{x\to e}\frac{f(e)-f(x)}{x-e}=-f'(e).
\]
Because
\[
f'(x)=\frac{1}{x},
\]
we get
\[
-f'(e)=-\frac{1}{e}.
\]
Q 4-A.
Let \(f(x)=\sin x\). Evaluate \( \displaystyle \lim_{x\to \pi}\frac{f(3x)-f(x)}{x-\pi} \).
A. \(-3\)
B. \(-2\)
C. \(0\)
D. \(2\)
Answer: B
As \(x\to\pi\), both numerator and denominator approach \(0\), so this is a \(0/0\) form. Differentiate numerator and denominator:
\[
\frac{d}{dx}\big(\sin(3x)-\sin x\big)=3\cos(3x)-\cos x.
\]
Therefore
\[
\lim_{x\to\pi}\frac{\sin(3x)-\sin x}{x-\pi}
=3\cos(3\pi)-\cos(\pi)=3(-1)-(-1)=-2.
\]
Q 4-B.
Let \(f(x)=\cos x\). Evaluate \( \displaystyle \lim_{x\to \tfrac{\pi}{2}}\frac{f(2x)-f(x)}{x-\tfrac{\pi}{2}} \).
A. \(-1\)
B. \(0\)
C. \(1\)
D. \(2\)
Answer: C
This is also \(0/0\). Differentiate the numerator:
\[
\frac{d}{dx}\big(\cos(2x)-\cos x\big)=-2\sin(2x)+\sin x.
\]
Then evaluate at \(x=\frac{\pi}{2}\):
\[
-2\sin(\pi)+\sin\left(\frac{\pi}{2}\right)=0+1=1.
\]
Q 5-A.
Let \( f(x)=-4\sin x+5e^{x} \). Find \( f'(0) \).
A. \(-9\)
B. \(-4\)
C. \(1\)
D. \(9\)
Answer: C
Differentiate term by term:
\[
f'(x)=-4\cos x+5e^x.
\]
Now substitute \(x=0\):
\[
f'(0)=-4\cos 0+5e^0=-4(1)+5(1)=1.
\]
Why do so many exponential limits turn into logarithms?
The natural logarithm appears because the derivative of \(a^x\) is \(a^x \ln a\). When the exponential expression is evaluated near zero in the standard form, the remaining constant is \(\ln a\).
How can I spot a derivative hidden inside a limit?
Look for patterns like \( \frac{f(a+h)-f(a)}{h} \) or \( \frac{f(x)-f(a)}{x-a} \). If the expression matches one of those forms, it is often asking for a derivative at a point.
What is the most common mistake with logarithmic difference quotients?
The most common mistake is missing a negative sign. If the numerator is reversed as \(f(a)-f(x)\), the limit becomes \(-f'(a)\), not \(f'(a)\).
Why are there A and B versions of the same type of question?
Parallel forms help students practice the same skill with different values. This is better for retention and reduces the temptation to memorize answer patterns without understanding the method.
