📝 Implicit differentiation examples (20 MCQs)

📖 Explanation: Differentiating implicitly gives $2x + 2y\,dy/dx = 0$ so $dy/dx = -x/y$. Setting $dy/dx = 0$ forces $x = 0$. Substituting into the original equation yields $y = \pm5$. Since both satisfy the condition, the y‑coordinate is not uniquely determined, hence option D.

📖 Explanation: Implicit differentiation gives $y + x\,dy/dx = 0$ so $dy/dx = -y/x$. As $x$ grows, the magnitude of $-y/x$ depends on the sign of $y$, which is not fixed by the information provided. Therefore the effect on $dy/dx$ cannot be determined from the given data.

📖 Explanation: Differentiating implicitly: $e^{xy}(y + x\,dy/dx) + dy/dx = 0$. At $x=0$ we have $e^{0}=1$ and the original equation gives $y = 2$. Substituting, $1\cdot (2 + 0) + dy/dx = 0$ leads to $dy/dx = -2$. Hence option A is correct.

📖 Explanation: Differentiating gives $\cos(x+y)(1 + dy/dx) = 1$. At $(0,0)$ we have $\cos0 = 1$, so $1 + dy/dx = 1$ and thus $dy/dx = 0$. The derivative is zero, making option C correct.

📖 Explanation: First differentiate: $3x^2 + 3y^2\,dy/dx = 6(y + x\,dy/dx)$. Setting $x=y$ yields $dy/dx = -1$. Differentiating again and simplifying using the original relation gives $d^2y/dx^2 = 1$ for all non‑zero points where $x=y$. Hence the statement is true.

📖 Explanation: With $y=1$ the first derivative simplifies to $dy/dx = -2x/(x^2+1)$. Computing the second derivative yields a rational expression whose sign depends on the value of $x$. Since no specific $x$ is given, the sign cannot be conclusively determined, so option D is appropriate.

📖 Explanation: Differentiating implicitly gives $2x + 2y\,dy/dx = 0$ leading to $dy/dx = -x/y$. Explicitly differentiating $y = \sqrt{r^2 - x^2}$ also yields $dy/dx = -x/\sqrt{r^2 - x^2}$, which is the same as $-x/y$. Thus both methods give the same derivative.

📖 Explanation: The equation mixes $x$ and $y$ in both exponential and linear terms, making it impractical to solve explicitly for $y$. Directly applying implicit differentiation to each term avoids solving for $y$ and yields a straightforward expression for $dy/dx$, so the implicit method is the most efficient.

📖 Explanation: When the first derivative is taken, the term $x^2 y$ contributes a factor of $dy/dx$. Differentiating again produces a $(dy/dx)^2$ piece from the derivative of the $y$ factor in both the $x^2 y$ term and the $y^3$ term. Hence both terms generate a squared derivative component.

📖 Explanation: Differentiating yields $-\sin(xy)(y + x\,dy/dx) = 1 + dy/dx$. At $(0,0)$ the sine term vanishes, leaving $0 = 1 + dy/dx$. Solving gives $dy/dx = -1$; therefore the derivative exists and equals $-1$.

📖 Explanation: First differentiate: $4x^3 + 4y^3\,dy/dx = 0$ giving $dy/dx = -x^3/y^3$. Differentiating again and substituting the original relation $x^4 + y^4 = 16$ simplifies the result to $-48x^2/y^7$. Thus option A is the correct formula.

📖 Explanation: Differentiating once yields an identity at $(1,1)$, leaving $dy/dx$ undetermined. Because the first derivative is not uniquely defined, the second derivative—and hence the curvature—cannot be computed from the given information alone. Therefore the curvature cannot be determined.

📖 Explanation: Differentiating implicitly gives $2x + 2y\,dy/dx = 0$. Solving for $dy/dx$ yields $-x/y$. This is the standard derivative for a circle centered at the origin.

📖 Explanation: The relation $x^{y}=2$ mixes $x$ and $y$ in a way that prevents solving for $y$ explicitly. Additionally, the derivative of a variable exponent requires the chain rule and logarithmic differentiation. Thus all listed reasons make implicit differentiation essential.

📖 Explanation: The equation $x^2 - y^2 = 4$ matches the standard form of a hyperbola $\frac{x^2}{4} - \frac{y^2}{4} = 1$. Hence the curve is a hyperbola.

📖 Explanation: The systematic approach is: (1) differentiate the original relation once, (2) solve the resulting equation for $dy/dx$, (3) differentiate that expression again with respect to $x$, and (4) substitute any previously found expressions to simplify. This yields the second derivative.

📖 Explanation: Differentiating gives $(y^2 - x)dy/dx = y - x^2$. Substituting $y = kx$ leads to $(k^2x^2 - x)dy/dx = kx - x^2$. As $x \to 0$, the dominant terms give $-x\,dy/dx \approx kx$, so $dy/dx \to -k$. Hence the derivative tends to $-k$.

📖 Explanation: From implicit differentiation, $dy/dx = -2xy/(x^2 + 3y^2)$. With $y>0$ fixed, as $x$ approaches zero from the positive side the numerator tends to zero while the denominator approaches $3y^2>0$. Consequently the fraction tends to 0, so the limit is 0.

📖 Explanation: Treating $y$ as a function of $x$, differentiate $F(x,y)=0$ to obtain $F_x + F_y\,dy/dx = 0$. Solving for $dy/dx$ gives $dy/dx = -F_x/F_y$. This compact formula is the cornerstone of implicit differentiation.

📖 Explanation: When a term involves a product such as $x\,y$ or $x^2 y$, the product rule $d(uv) = u\,dv + v\,du$ is used, remembering that $y$ depends on $x$. This rule is essential for correctly handling mixed terms in implicit equations.