<p>Let \(x^3 - 2x^2y^2 + 5x + y - 5 = 0\) and at \(x = 1\), \(y = 1\). Then \(\dfrac{dy}{dx}\) at \(y = 1\) is</p>
Step-by-Step Solution
Key Concept: Use implicit differentiation on the given equation, then substitute the point (1,1) to find dy/dx. The key is recognizing that all terms must be differentiated with respect to x, treating y as a function of x.
<p><strong>Step 1: Differentiate both sides with respect to x</strong></p><p>Given: x³ - 2x²y² + 5x + y - 5 = 0</p><p>Differentiating implicitly:</p><p>3x² - [4xy² + 2x²(2y)(dy/dx)] + 5 + dy/dx = 0</p><p>3x² - 4xy² - 4x²y(dy/dx) + 5 + dy/dx = 0</p><p><strong>Step 2: Rearrange to isolate dy/dx</strong></p><p>dy/dx(1 - 4x²y) = 4xy² - 3x² - 5</p><p>dy/dx = (4xy² - 3x² - 5)/(1 - 4x²y)</p><p><strong>Step 3: Substitute the point (x=1, y=1)</strong></p><p>dy/dx = (4(1)(1)² - 3(1)² - 5)/(1 - 4(1)²(1))</p><p>dy/dx = (4 - 3 - 5)/(1 - 4)</p><p>dy/dx = (-4)/(-3)</p><p>dy/dx = 4/3</p><p>∴ Answer: A</p>
Correct Answer: A