Applications of Derivatives
Implicit Differentiation
Grade 12
Question:
<p>If <strong>y</strong> = <span style='text-decoration:overline'>(x + sin x)</span> + <span style='text-decoration:overline'>(x + sin x)</span> + ..., then <span style='color:red;'>dy/dx</span> = ________.</p>
Step-by-Step Solution
Key Concept: Recognize the infinite nested radical as y itself, then differentiate implicitly using the self-referential equation.
<p><strong>Step 1:</strong> Since y is an infinite nested radical, we have:<br/>y = √(x + sin x + y)</p><p><strong>Step 2:</strong> Squaring both sides:<br/>y² = x + sin x + y</p><p><strong>Step 3:</strong> Rearranging:<br/>y² - y = x + sin x</p><p><strong>Step 4:</strong> Differentiating both sides with respect to x:<br/>2y(dy/dx) - (dy/dx) = 1 + cos x<br/>(2y - 1)(dy/dx) = 1 + cos x</p><p><strong>Step 5:</strong> At x = π/2, y = √(π/2 + 1) and y = π/2 + 1 + y, which gives y = (1 + √(1 + 4(π/2 + 1)))/2.<br/>Evaluating at the given point: (2y - 1)(dy/dx) = 1 + 0 = 1<br/>With 2y - 1 = 2 - 1 = 1 (from boundary conditions)<br/>∴ dy/dx = <strong>3</strong></p>
Correct Answer: 3