Differentiation Practice Questions
AP (Advanced Placement) · AP Calculus BC · 147 free MCQs with instant results and detailed explanations.
147
Total
35
Easy
75
Medium
37
Hard
Start Practicing Differentiation
Take a timed quiz or customize your practice session
Sample Questions from Differentiation
Here are 10 sample questions. Start a quiz to get randomized questions with scoring.
Q1
Easy
What is the slope of the tangent line to the curve y = sin(x) at x = ฯ/4?
A.
1/โ2
B.
โ2
C.
0
D.
1
Show Answer & Explanation
Correct Answer: A
To find the slope of the tangent line at x = ฯ/4, we compute the derivative y' = cos(x). Then, substituting x = ฯ/4 gives y'(ฯ/4) = cos(ฯ/4) = 1/โ2, which is the slope of the tangent line.
Q2
Easy
What is the derivative of the function f(x) = 3x^4 - 5x^2 + 2?
A.
12x^3 - 10x
B.
3x^3 - 5
C.
9x^2 - 5
D.
6x^3 + 2
Show Answer & Explanation
Correct Answer: A
The derivative of f(x) is found using the power rule. For 3x^4, the derivative is 12x^3, and for -5x^2, it is -10x. Thus, the derivative is 12x^3 - 10x.
Q3
Easy
If f(x) = x^2 + 4x + 4, what is the slope of the tangent line at x = 1?
A.
6
B.
5
C.
4
D.
2
Show Answer & Explanation
Correct Answer: A
First, find the derivative f'(x) = 2x + 4. Plugging in x = 1 gives f'(1) = 2(1) + 4 = 6, which is the slope of the tangent line.
Q4
Medium
If f(x) = sin(x^2), what is the value of f'(ฯ) using the chain rule?
A.
0
B.
-2ฯ
C.
2ฯ
D.
ฯ
Show Answer & Explanation
Correct Answer: C
Using the chain rule, f'(x) = cos(x^2) * 2x. Therefore, f'(ฯ) = cos(ฯ^2) * 2ฯ. Since cos(ฯ^2) is not zero, the term 2ฯ remains, making the answer C.
Q5
Medium
What is the second derivative of the function f(x) = e^(2x) * cos(x)?
A.
e^(2x)(2cos(x) - sin(x))
B.
e^(2x)(2sin(x) + cos(x))
C.
2e^(2x)cos(x) - e^(2x)sin(x)
D.
e^(2x)(cos(x) - 2sin(x))
Show Answer & Explanation
Correct Answer: A
To find the second derivative, use the product and chain rules. The first derivative is f'(x) = e^(2x)(2cos(x) - sin(x)). Differentiating again gives the second derivative as e^(2x)(2cos(x) - sin(x)). Thus, the answer is A.
Q6
Medium
Determine the critical points of the function f(x) = x^4 - 8x^2 + 16.
A.
x = ยฑ2
B.
x = 0, ยฑ2
C.
x = ยฑ4
D.
x = 0
Show Answer & Explanation
Correct Answer: B
To find critical points, set the first derivative f'(x) = 0. f'(x) = 4x^3 - 16x. Factoring gives 4x(x^2 - 4) = 0, which leads us to x = 0, ยฑ2. Therefore, the correct answer is B.
Q7
Medium
For the function f(x) = ln(x^2 + 1), what is f''(x) when x = 1?
A.
0
B.
1/2
C.
-1/2
D.
1
Show Answer & Explanation
Correct Answer: B
First, find f'(x) = (2x)/(x^2 + 1). Then, differentiate again to find f''(x). Evaluating f''(1) yields 1/2. Hence, the correct answer is B.
Q8
Hard
Let f(x) = x^3 - 6x^2 + 9x. What is the x-coordinate of the point where the function has a local minimum?
A.
1
B.
2
C.
3
D.
4
Show Answer & Explanation
Correct Answer: B
To find the local minimum, we first calculate the derivative f'(x) = 3x^2 - 12x + 9. Setting f'(x) = 0 gives us the critical points. Solving 3(x^2 - 4x + 3) = 0 leads to (x - 1)(x - 3) = 0, hence x = 1 or x = 3. To determine the nature of these points, we can use the second derivative test. f''(x) = 6x - 12. Evaluating f''(2) gives us 0, indicating a local minimum at x = 2.
Q9
Hard
Consider the function g(t) = t^4 - 8t^3 + 18t^2. Which of the following statements is true regarding the behavior of g(t) at t = 0?
A.
g(t) has a local maximum at t = 0.
B.
g(t) has a local minimum at t = 0.
C.
g(t) has an inflection point at t = 0.
D.
The value of g(t) at t = 0 is negative.
Show Answer & Explanation
Correct Answer: B
To analyze g(t) at t = 0, we first find g'(t) = 4t^3 - 24t^2 + 36t. Setting g'(0) = 0 shows that t = 0 is a critical point. Evaluating the second derivative g''(t) = 12t^2 - 48t + 36, gives g''(0) = 36, which is positive. Therefore, g(t) has a local minimum at t = 0.
Q10
Hard
Evaluate the limit: lim (x -> 0) [(sin(5x) - 5x) / (x^3)]. What does this limit equal?
A.
0
B.
-5/3
C.
25/6
D.
5/6
Show Answer & Explanation
Correct Answer: C
Using L'Hรดpital's rule, since it is of the form 0/0, differentiate the numerator and denominator three times. The third derivative of sin(5x) at x=0 is 125 while the derivative of x^3 is 6. Therefore, the limit evaluates to 125/6, simplifying to 25/6.
Showing 10 of 147 questions. Start a quiz to practice all questions with scoring and timer.
Practice All 147 Questions →Other AP (Advanced Placement) Subjects
Differentiation โ AP (Advanced Placement) AP Calculus BC Practice Questions Online
This page contains 147 practice MCQs for the chapter Differentiation in AP (Advanced Placement) AP Calculus BC. The questions are organized by difficulty โ 35 easy, 75 medium, 37 hard โ so you can choose the right level for your preparation.
Every question includes a detailed explanation to help you understand the concept, not just memorize answers. Take a timed quiz to simulate exam conditions, or practice at your own pace with no time limit.