Limits, Continuity & Differentiability
Differentiation of composite functions
Grade 12
Question:
<p><strong>274.</strong> \(y = \cos^{-1}\!\left(\log_2 2^{\ln e^{\sin^{-1}\sin x}}\right)\). For \(y\) as defined above, the value of \(\frac{dy}{dx}\) at \(x = \frac{\pi}{4}\) is:</p>
<p>(a) 0</p>
<p>(b) \(\dfrac{1}{2}\)</p>
<p>(c) \(\dfrac{1}{\sqrt{2}}\)</p>
<p>(d) 1</p>
Step-by-Step Solution
Key Concept: Simplify the nested functions step-by-step: recognize that log₂(2^a) = a, and sin⁻¹(sin x) = x for x ∈ [-π/2, π/2]. Then differentiate cos⁻¹(u) using the chain rule with du/dx.
<p><strong>Step 1:</strong> Simplify the argument inside cos⁻¹.</p><p>y = cos⁻¹(log₂ 2^(ln e^(sin⁻¹ sin x)))</p><p>Since ln e^(a) = a, we have:</p><p>y = cos⁻¹(log₂ 2^(sin⁻¹ sin x))</p><p><strong>Step 2:</strong> Apply logarithm property log₂ 2^(a) = a.</p><p>y = cos⁻¹(sin⁻¹ sin x)</p><p><strong>Step 3:</strong> For x = π/4 ∈ [-π/2, π/2], we have sin⁻¹(sin x) = x.</p><p>y = cos⁻¹(x)</p><p><strong>Step 4:</strong> Differentiate cos⁻¹(x) with respect to x.</p><p>dy/dx = -1/√(1 - x²)</p><p><strong>Step 5:</strong> Evaluate at x = π/4.</p><p>dy/dx|ₓ₌π/₄ = -1/√(1 - (π/4)²) = -1/√(1 - π²/16)</p><p>= -1/√((16 - π²)/16) = -4/√(16 - π²)</p><p>∴ Answer: D</p>
Correct Answer: D