Question Bank • QB_EM_ADV_034
MCQ • 1 Mark
Section 1: Engineering Mathematics
Calculus
Subtopic: Total differentiation
Question Statement:
If an implicit equation $f(x, y) = 0$ defines $y$ as a differentiable function of $x$, then the total derivative $\frac{dy}{dx}$ is given by:
Options:
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(A) $-\frac{\partial f / \partial x}{\partial f / \partial y}$
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(B) $\frac{\partial f / \partial x}{\partial f / \partial y}$
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(C) $-\frac{\partial f / \partial y}{\partial f / \partial x}$
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(D) $\frac{\partial f / \partial y}{\partial f / \partial x}$
Step-by-Step Solution & Explanation
Taking the total differential of $f(x, y) = 0$:
$df = \frac{\partial f}{\partial x} dx + \frac{\partial f}{\partial y} dy = 0$
Solving for $\frac{dy}{dx}$:
$\frac{\partial f}{\partial y} dy = -\frac{\partial f}{\partial x} dx \implies \frac{dy}{dx} = -\frac{\partial f / \partial x}{\partial f / \partial y}$
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