Limits, Continuity & Differentiability
Differentiation
Grade 12
Question:
<p>Let \(y = \log \sin(x^2)\), \(0 < x \leq \dfrac{\pi}{2}\). The value of \(\dfrac{dy}{dx}\) at \(x = \dfrac{\sqrt{\pi}}{2}\) is</p>
<p>(a) 0</p>
<p>(b) 1</p>
<p>(c) \(\pi/4\)</p>
<p>(d) \(\sqrt{\pi}\)</p>
Step-by-Step Solution
Key Concept: Use logarithmic differentiation combined with chain rule. The derivative of log(sin(x²)) requires careful application of chain rule: dy/dx = (1/sin(x²)) · cos(x²) · 2x = 2x·cot(x²).
<p><strong>Step 1:</strong> Start with y = log sin(x²)</p><p><strong>Step 2:</strong> Differentiate using chain rule: dy/dx = (1/sin(x²)) · d/dx[sin(x²)]</p><p><strong>Step 3:</strong> Apply chain rule to sin(x²): d/dx[sin(x²)] = cos(x²) · 2x</p><p><strong>Step 4:</strong> Combine: dy/dx = (1/sin(x²)) · cos(x²) · 2x = 2x · cos(x²)/sin(x²)</p><p><strong>Step 5:</strong> Simplify using cot identity: dy/dx = 2x·cot(x²)</p><p>∴ Answer: D</p>
Correct Answer: D