CALCULUS 1 IMPLICIT DIFFERENTIATION
MASTER QUESTION BANK ( MCQS WITH
RATIONALES)
This premium study resource contains a comprehensive
bank of multiple-choice practice questions tracking
advanced implicit differentiation applications. Every single
problem is explicitly mapped with its corresponding answer
key and a rigorous, step-by-step mathematical rationale
embedded directly inline for maximum efficiency. It serves as
an elite exam preparation tool tailored for Calculus I students
looking to master complex curves, chain rules, and
coordinate tangent analysis.
Question 1
Find
dydxd y over d x end-fraction
𝑑𝑦𝑑𝑥
for the equation x² + y² = 36.
A)
𝑑𝑦𝑑𝑥
=
,−𝑦𝑥
B)
𝑑𝑦𝑑𝑥
=
−𝑥𝑦
C)
𝑑𝑦𝑑𝑥
=𝑥𝑦
D)
𝑑𝑦𝑑𝑥
=
−2𝑥
Answer: B
Rationale: Differentiating both sides with respect to x yields
2𝑥
+2𝑦
𝑑𝑦𝑑𝑥
=0
. Isolating
,dydxd y over d x end-fraction
𝑑𝑦𝑑𝑥
gives
2𝑦
𝑑𝑦𝑑𝑥
=
−2𝑥
, which simplifies directly to
−xynegative x over y end-fraction
−𝑥𝑦
.
Question 2
Find the derivative
dydxd y over d x end-fraction
𝑑𝑦𝑑𝑥
of the curve y³ + x³ = 8.
A)
, −x2y2negative the fraction with numerator x squared and
denominator y squared end-fraction
−𝑥2𝑦2
B)
−y2x2negative the fraction with numerator y squared and
denominator x squared end-fraction
−𝑦2𝑥2
C)
x2y2the fraction with numerator x squared and denominator y
squared end-fraction
𝑥2𝑦2
D) -3x²
Answer: A
Rationale: Differentiating both sides implicitly gives
3𝑦2
𝑑𝑦𝑑𝑥
+3𝑥2
=0
. Subtracting 3x² from both sides gives
MASTER QUESTION BANK ( MCQS WITH
RATIONALES)
This premium study resource contains a comprehensive
bank of multiple-choice practice questions tracking
advanced implicit differentiation applications. Every single
problem is explicitly mapped with its corresponding answer
key and a rigorous, step-by-step mathematical rationale
embedded directly inline for maximum efficiency. It serves as
an elite exam preparation tool tailored for Calculus I students
looking to master complex curves, chain rules, and
coordinate tangent analysis.
Question 1
Find
dydxd y over d x end-fraction
𝑑𝑦𝑑𝑥
for the equation x² + y² = 36.
A)
𝑑𝑦𝑑𝑥
=
,−𝑦𝑥
B)
𝑑𝑦𝑑𝑥
=
−𝑥𝑦
C)
𝑑𝑦𝑑𝑥
=𝑥𝑦
D)
𝑑𝑦𝑑𝑥
=
−2𝑥
Answer: B
Rationale: Differentiating both sides with respect to x yields
2𝑥
+2𝑦
𝑑𝑦𝑑𝑥
=0
. Isolating
,dydxd y over d x end-fraction
𝑑𝑦𝑑𝑥
gives
2𝑦
𝑑𝑦𝑑𝑥
=
−2𝑥
, which simplifies directly to
−xynegative x over y end-fraction
−𝑥𝑦
.
Question 2
Find the derivative
dydxd y over d x end-fraction
𝑑𝑦𝑑𝑥
of the curve y³ + x³ = 8.
A)
, −x2y2negative the fraction with numerator x squared and
denominator y squared end-fraction
−𝑥2𝑦2
B)
−y2x2negative the fraction with numerator y squared and
denominator x squared end-fraction
−𝑦2𝑥2
C)
x2y2the fraction with numerator x squared and denominator y
squared end-fraction
𝑥2𝑦2
D) -3x²
Answer: A
Rationale: Differentiating both sides implicitly gives
3𝑦2
𝑑𝑦𝑑𝑥
+3𝑥2
=0
. Subtracting 3x² from both sides gives
