← Back to 14. Partial Derivatives Calculus

📝 Implicit differentiation with partial derivatives (14 MCQs)

📖 From Calculus • 14. Partial Derivatives Calculus • 14 questions available

←Previous Topic:Partial derivative chain rule📝 Implicit differentiation with partial derivativesDirectional Derivatives and Gradients:Next Topic→

What is Implicit differentiation with partial derivatives?

Definition:
Using chain rule on identity $F(x,y,z(x,y))=0$ to derive $z_x = -F_x/F_z$ and $z_y = -F_y/F_z$ without explicit solution.

Example:
For $x^3 + y^3 + z^3 = 3xyz$ , $\frac{\partial z}{\partial x} = -\frac{3x^2-3yz}{3z^2-3xy} = \frac{yz-x^2}{z^2-xy}$ .

Reason:
Enables derivative computation for implicitly defined surfaces common in geometry and physics where explicit isolation is impossible or impractical.

Easy

Medium

Hard

📝 All Implicit differentiation with partial derivatives MCQs

Q1. A thermodynamic surface is defined by $F(P,V,T) = PV - nRT + aP^2 = 0$ . When analyzing the adiabatic compressibility, one must find $\left(\frac{\partial V}{\partial P}\right)_T$ . Which expression correctly represents this partial derivative using implicit differentiation?

-\frac{V + 2aP}{P}

✅

\frac{P}{V + 2aP}

-\frac{nR}{V + 2aP}

\frac{V - nRT}{P + 2aP}

💡 Difficulty: hard | ✅ Correct: A

📖 Explanation: This question requires applying the implicit function theorem to a multivariable physical model. Students must identify that T is held constant, treating it as a parameter, and correctly apply the negative ratio of partial derivatives formula $-F_P / F_V$ , avoiding sign errors common in thermodynamic relations.

Q2. Consider the level curve $x^3 + y^3 = 3xy$ . At the point where the tangent line is vertical, what condition must be satisfied by the coordinates $(x,y)$ ?

y = x^2

and

x^3 + y^3 = 3xy

x = y^2

and

x^3 + y^3 = 3xy

✅

y = 0

and

x = 0

x^2 = y

and

3x^2 - 3y = 0

💡 Difficulty: medium | ✅ Correct: B

📖 Explanation: Vertical tangents occur when $dy/dx$ is undefined, meaning the denominator of the implicit derivative equals zero while the numerator is non-zero. For this folium, setting $\partial F/\partial y = 0$ yields $3y^2 - 3x = 0$ , leading to $x = y^2$ . This tests conceptual understanding of geometric singularities rather than rote computation.

Q3. A student attempts to find $\frac{dy}{dx}$ for $\sin(xy) = x + y$ and obtains $\frac{1 - y\cos(xy)}{x\cos(xy) - 1}$ . Identify the specific error in their reasoning.

A.They forgot to apply the chain rule to the sine function.

B.They incorrectly assigned the sign when moving terms across the equality. ✅

C.They treated

y

as a constant during the differentiation of

xy

D.They differentiated the right side as zero instead of one.

💡 Difficulty: medium | ✅ Correct: B

📖 Explanation: The correct derivative involves collecting $dy/dx$ terms. The student's numerator and denominator have swapped signs relative to the standard form $-(F_x/F_y)$ . This error analysis question targets the common algebraic misconception regarding sign distribution when isolating the derivative term after implicit differentiation.

Q4. Given the contour plot of a function $z = f(x,y)$ where level curves are densely packed near point A and sparse near point B, if both points lie on the same implicit curve $f(x,y)=c$ , how does the magnitude of $dy/dx$ compare?

A.The magnitude is necessarily larger at A because gradients are steeper.

B.The magnitude depends entirely on the orientation of the gradient vector relative to the axes, not just density. ✅

C.The magnitude is necessarily smaller at A due to higher constraint stiffness.

D.The magnitudes are identical because they share the same level value c.

💡 Difficulty: hard | ✅ Correct: B

📖 Explanation: This graph-based question challenges the misconception that contour density directly dictates slope magnitude along a curve. While density indicates gradient magnitude, the implicit slope $-f_x/f_y$ depends on the ratio of components. Steep gradients can still yield shallow slopes if both partials scale similarly, requiring spatial reasoning.

Q5. For the equation $e^{xy} + \ln(x+y) = 5$ , determine the value of $\frac{dy}{dx}$ at the point $(1, e-1)$ .

-\frac{e(e-1) + \frac{1}{e}}{e + \frac{1}{e}}

✅

\frac{e(e-1) + \frac{1}{e}}{e + \frac{1}{e}}

-\frac{e + \frac{1}{e}}{e(e-1) + \frac{1}{e}}

D.Zero, because the function is constant.

💡 Difficulty: medium | ✅ Correct: A

📖 Explanation: This application problem requires evaluating transcendental derivatives at a specific coordinate. Students must correctly apply product and chain rules to both exponential and logarithmic terms simultaneously. The complexity lies in accurate substitution and simplification, testing procedural fluency combined with careful arithmetic in a multi-step context.

Q6. Why is it mathematically invalid to use implicit differentiation to find $dy/dx$ at a point where both $F_x = 0$ and $F_y = 0$ simultaneously?

A.Because the function is discontinuous at that point.

B.Because the Implicit Function Theorem guarantees local existence of a differentiable function only when

F_y \neq 0

. ✅

C.Because the derivative becomes infinite at all such points.

D.Because such points always represent global maxima or minima.

💡 Difficulty: hard | ✅ Correct: B

📖 Explanation: This conceptual question addresses the theoretical foundation of the method. When both partials vanish, the point is singular, and the curve may self-intersect or have a cusp. The derivative is undefined not merely infinite, making standard implicit differentiation inapplicable. Understanding this limitation prevents misuse in advanced modeling scenarios.

Q7. Compare finding $dy/dx$ for $x^2 + y^2 = 25$ via explicit solving versus implicit differentiation. In which scenario does implicit differentiation offer a distinct computational advantage?

A.When calculating the derivative at points where

y > 0

only.

B.When the relationship cannot be easily solved for

y

as a single-valued function of

x

. ✅

C.When higher-order derivatives are required exclusively.

D.Never; explicit differentiation is always computationally superior.

💡 Difficulty: easy | ✅ Correct: B

📖 Explanation: This mixed-concept question evaluates method selection. Explicit solving yields two branches requiring piecewise handling, while implicit differentiation provides a unified expression valid for all regular points. Recognizing when algebraic isolation is impractical or ambiguous is crucial for efficient problem-solving in complex multivariable contexts.

Q8. If $z$ is defined implicitly by $xz + yz^2 = 10$ , which expression gives $\frac{\partial z}{\partial x}$ holding $y$ constant?

-\frac{z}{x + 2yz}

✅

-\frac{x + 2yz}{z}

\frac{z}{x + yz}

-\frac{z + yz^2}{x}

💡 Difficulty: medium | ✅ Correct: A

📖 Explanation: Students must distinguish between total and partial implicit differentiation. Holding $y$ constant means treating it as a parameter, not a variable. The correct application uses $-F_x/F_z$ where $F_x = z$ and $F_z = x + 2yz$ . Confusing this with total derivative setups is a frequent misconception tested here.

Q9. An engineer models a constraint $g(x,y) = x^4 + y^4 - 4x^2y^2 = 0$ . Near the origin, numerical solvers fail to converge for $dy/dx$ . What is the most likely mathematical reason?

A.The origin is a regular point but solver tolerance is too loose.

B.The origin satisfies

g_x = g_y = 0

, creating a singularity where the implicit derivative is undefined. ✅

C.The function is not continuous at the origin.

D.The constraint defines no real solutions near the origin.

💡 Difficulty: hard | ✅ Correct: B

📖 Explanation: This scenario-based error analysis links computational failure to theoretical pathology. The quartic curve has a tacnode at the origin where both partials vanish. Recognizing that numerical instability signals a violation of IFT conditions helps diagnose modeling issues beyond mere coding errors, integrating theory with practical application.

Q10. Given $F(x,y,z) = 0$ defines $z$ implicitly as a function of $x$ and $y$ , which identity must hold for the mixed partial $\frac{\partial^2 z}{\partial x \partial y}$ ?

A.It always equals

\frac{\partial^2 z}{\partial y \partial x}

regardless of continuity.

B.It can be expressed solely in terms of first and second partials of

F

evaluated at the point. ✅

C.It is always zero for homogeneous functions.

D.It requires solving a system of three linear equations.

💡 Difficulty: hard | ✅ Correct: B

📖 Explanation: This Olympiad-style question probes deep structural knowledge. Computing second-order implicit derivatives involves quotient rule applications on $-F_x/F_z$ , generating expressions involving $F_{xx}, F_{xy}, F_{xz}, F_{zz}$ , etc. Knowing this dependency structure without full derivation demonstrates mastery of the underlying calculus machinery beyond mechanical computation.

Q11. A student claims that for any implicit curve $F(x,y)=c$ , the gradient vector $\nabla F$ is parallel to the tangent vector. Evaluate this statement.

A.True, because gradients always point in the direction of motion.

B.False, the gradient is orthogonal to the level curve, hence perpendicular to the tangent vector. ✅

C.True, but only for circles and ellipses.

D.False, the gradient is parallel only when

c=0

💡 Difficulty: easy | ✅ Correct: B

📖 Explanation: This direct recall question reinforces fundamental geometry. Misconceptions about gradient direction persist despite instruction. Clarifying that $\nabla F \perp \text{tangent}$ underpins why $dy/dx = -F_x/F_y$ : the slope comes from orthogonality conditions. Establishing this baseline supports higher-order reasoning in subsequent problems.

Q12. In economics, an indifference curve $U(x,y) = k$ exhibits diminishing marginal rate of substitution. If $U_{xy} > 0$ and $U_x, U_y > 0$ , what can be inferred about $d^2y/dx^2$ along the curve?

A.It must be positive everywhere.

B.Its sign depends on the relative magnitudes of second partials and cannot be determined from

U_{xy}>0

alone. ✅

C.It must be negative, confirming convexity.

D.It is identically zero for Cobb-Douglas utilities.

💡 Difficulty: hard | ✅ Correct: B

📖 Explanation: This mixed-concept application connects implicit differentiation to economic theory. Diminishing MRS implies convex indifference curves ( $d^2y/dx^2 > 0$ in standard orientation), but the sign depends on a complex combination of $U_{xx}, U_{yy}, U_{xy}$ . Assuming positivity from cross-partial alone reflects superficial understanding, testing nuanced interpretation.

Q13. When differentiating $x^y = y^x$ implicitly, which step is essential before applying standard rules?

A.Taking natural logarithms of both sides to convert exponents to products. ✅

B.Differentiating directly using power rule treating base and exponent as variables.

C.Substituting

y = x

first since it satisfies the equation.

D.Converting to parametric form using

t = y/x

💡 Difficulty: medium | ✅ Correct: A

📖 Explanation: This application question targets a specialized technique. Direct differentiation fails because neither base nor exponent is constant. Logarithmic transformation converts the expression to $y\ln x = x\ln y$ , enabling product rule application. Recognizing when standard rules are insufficient and selecting appropriate preprocessing is critical for handling variable-exponent forms.

Q14. Suppose $F(x,y) = 0$ defines $y$ implicitly near $(a,b)$ with $F_y(a,b) \neq 0$ . If a perturbation changes the constraint to $F(x,y) = \epsilon$ for small $\epsilon$ , how does the new derivative $dy/dx$ relate to the original?

A.It remains exactly unchanged since derivatives are local properties.

B.It changes by approximately

\epsilon \cdot \frac{\partial}{\partial c}\left(-\frac{F_x}{F_y}\right)

evaluated at

c=0

. ✅

C.It becomes undefined for any nonzero

\epsilon

D.It shifts by

\epsilon / F_y

regardless of

F_x

💡 Difficulty: hard | ✅ Correct: B

📖 Explanation: This challenging question extends implicit differentiation to sensitivity analysis. The derivative itself depends on the level set value through the partial derivatives' dependence on position. First-order perturbation theory shows the change involves differentiating the implicit derivative expression with respect to the constraint parameter, testing advanced conceptual integration beyond static computation.

🔗 Related Topics (MCQs)

📝 Functions of Two or More Variables MCQs 📝 Multivariable function notation and terminology MCQs 📝 Functions of two variables from tables MCQs 📝 Level curves and contour plots MCQs 📝 Graphing functions of two variables MCQs

←Previous Topic:Partial derivative chain rule📝 Implicit differentiation with partial derivativesDirectional Derivatives and Gradients:Next Topic→