Limits, Continuity & Differentiability
Differentiation of composite / inverse functions
Grade 12
Question:
<p>\(y = \cos^{-1}\!\left(\log_2 2^{\ln e^{\sin^{-1}\sin x}}\right)\). For \(y\) as defined above, the value of \(\dfrac{dy}{dx}\) at \(x = \dfrac{\pi}{4}\) 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 composition by recognizing that $\log_2 2^{a} = a$ and $\sin^{-1}(\sin x) = x$ for $x \in [-\pi/2, \pi/2]$, reducing the problem to differentiating a simple inverse cosine function.
<p><strong>Step 1:</strong> Simplify the innermost expression: $\sin^{-1}(\sin x) = x$ for $x = \frac{\pi}{4} \in [-\frac{\pi}{2}, \frac{\pi}{2}]$</p><p><strong>Step 2:</strong> Apply the natural logarithm property: $\ln(e^{\sin^{-1}\sin x}) = \ln(e^x) = x$</p><p><strong>Step 3:</strong> Apply the logarithm base conversion: $\log_2(2^x) = x$</p><p><strong>Step 4:</strong> Therefore, $y = \cos^{-1}(x)$</p><p><strong>Step 5:</strong> Differentiate: $\frac{dy}{dx} = -\frac{1}{\sqrt{1-x^2}}$</p><p><strong>Step 6:</strong> Evaluate at $x = \frac{\pi}{4}$: $\frac{dy}{dx}\bigg|_{x=\pi/4} = -\frac{1}{\sqrt{1-(\frac{\pi}{4})^2}}$</p><p>∴ Answer: A</p>
Correct Answer: A