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Differentiation Composite and Implicit Practice Questions

AP (Advanced Placement) · AP Calculus AB · 143 free MCQs with instant results and detailed explanations.

143
Total
50
Easy
73
Medium
20
Hard

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Sample Questions from Differentiation Composite and Implicit

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Q1
Easy
If y = sin(x^2), what is dy/dx using implicit differentiation?
A. 2x cos(x^2)
B. cos(x^2)
C. 2x sin(x^2)
D. -2x sin(x^2)
Show Answer & Explanation
Correct Answer: A
Using the chain rule, dy/dx = cos(x^2) * d(x^2)/dx where d(x^2)/dx = 2x. Thus, dy/dx = 2x cos(x^2).
Q2
Easy
What is the derivative of the function f(x) = (3x^2 + 2x)^4 with respect to x?
A. 12x(3x^2 + 2x)^3
B. 4(3x^2 + 2x)^3(6x + 2)
C. 6(3x^2 + 2x)^4
D. 3(3x^2 + 2x)^4
Show Answer & Explanation
Correct Answer: B
Using the chain rule, the derivative of the outside function (4u^3) is multiplied by the derivative of the inside function (6x + 2), where u = 3x^2 + 2x.
Q3
Easy
If y = x^2 + 3x + 2 and x = 1, what is dy/dx using implicit differentiation?
A. 5
B. 6
C. 4
D. 3
Show Answer & Explanation
Correct Answer: A
Differentiating y with respect to x gives dy/dx = 2x + 3. Substituting x = 1 results in dy/dx = 2(1) + 3 = 5.
Q4
Medium
If y = sin(2x) + cos(x), what is the derivative dy/dx?
A. 2cos(2x) - sin(x)
B. 2sin(2x) - cos(x)
C. 2cos(2x) + sin(x)
D. 2cos(x) - sin(2x)
Show Answer & Explanation
Correct Answer: A
Using the chain and product rules, the derivative of sin(2x) is 2cos(2x) and the derivative of cos(x) is -sin(x). Thus, dy/dx = 2cos(2x) - sin(x).
Q5
Medium
If x^2 + y^2 = 25, use implicit differentiation to find dy/dx.
A. -x/y
B. x/y
C. y/x
D. -y/x
Show Answer & Explanation
Correct Answer: A
Differentiating both sides with respect to x gives 2x + 2y(dy/dx) = 0. Solving for dy/dx yields dy/dx = -x/y.
Q6
Medium
What is the derivative of y = ln(x^2 + 1) with respect to x?
A. 1/(x^2 + 1)
B. 2x/(x^2 + 1)
C. 2/(x^2 + 1)
D. 1/x
Show Answer & Explanation
Correct Answer: B
Using the chain rule, the derivative of ln(u) is 1/u * du/dx. Here, u = x^2 + 1, so dy/dx = 1/(x^2 + 1) * 2x = 2x/(x^2 + 1).
Q7
Medium
If f(x) = e^(3x)sin(2x), what is f'(x)?
A. e^(3x)(3sin(2x) + 2cos(2x))
B. 3e^(3x)sin(2x) + 2e^(3x)cos(2x)
C. e^(3x)(3sin(2x) - 2cos(2x))
D. e^(3x)(6sin(2x) + 3cos(2x))
Show Answer & Explanation
Correct Answer: B
Using the product rule, the derivative is f'(x) = e^(3x)(3sin(2x) + 2cos(2x)). Hence the correct option corresponds to the full differentiation.
Q8
Hard
Consider the implicit function defined by x^2 + y^2 = 25. What is the value of dy/dx at the point (3, 4)?
A. -3/4
B. 4/3
C. 3/4
D. -4/3
Show Answer & Explanation
Correct Answer: A
To find dy/dx implicitly, we differentiate both sides with respect to x: 2x + 2y(dy/dx) = 0. Solving for dy/dx gives dy/dx = -x/y. At the point (3, 4), substituting x = 3 and y = 4 yields dy/dx = -3/4.
Q9
Hard
Given the function y = sin(ln(x^2 + 1)), find the derivative dy/dx using the chain rule.
A. frac{2x cos(ln(x^2 + 1))}{x^2 + 1}
B. frac{2x sin(ln(x^2 + 1))}{x^2 + 1}
C. frac{cos(ln(x^2 + 1))}{x^2 + 1}
D. frac{2x^2 cos(ln(x^2 + 1))}{(x^2 + 1)^2}
Show Answer & Explanation
Correct Answer: A
Using the chain rule, the derivative of sin(u) is cos(u) * du/dx. Here, u = ln(x^2 + 1), with du/dx = (2x)/(x^2 + 1). Thus, dy/dx = cos(ln(x^2 + 1)) * (2x)/(x^2 + 1), simplifying to (2x cos(ln(x^2 + 1)))/(x^2 + 1).
Q10
Hard
If x^2 + y^2 = 25, find dy/dx using implicit differentiation at the point (3,4).
A. -frac{3}{4}
B. -frac{4}{3}
C. frac{4}{3}
D. frac{3}{4}
Show Answer & Explanation
Correct Answer: B
Using implicit differentiation on x^2 + y^2 = 25, we differentiate both sides to get 2x + 2y(dy/dx) = 0. Rearranging gives dy/dx = -x/y. Substituting (3, 4) gives dy/dx = -3/4, or -4/3 when evaluated at (3, 4).

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Differentiation Composite and Implicit — AP (Advanced Placement) AP Calculus AB Practice Questions Online

This page contains 143 practice MCQs for the chapter Differentiation Composite and Implicit in AP (Advanced Placement) AP Calculus AB. The questions are organized by difficulty — 50 easy, 73 medium, 20 hard — 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.