Limits, Continuity & Differentiability
Logarithmic Differentiation
Grade 12
Question:
<p>If <span class='math'>y = (\cos x)^{(\cos x)^{(\cos x)^{\cdots \infty}}}</span>, then <span class='math'>\frac{dy}{dx}</span> is equal to</p>
<p>(a) <span class='math'>\frac{y\tan x}{1-y\log(\cos x)}</span></p>
<p>(b) <span class='math'>\frac{y^2\tan x}{1-y\log(\cos x)}</span></p>
<p>(c) <span class='math'>-y\tan x</span></p>
<p>(d) <span class='math'>\frac{y^2\tan x}{\log(\cos x)}</span></p>
Step-by-Step Solution
Key Concept: Recognize self-similar infinite power and use logarithmic differentiation on the resulting equation.
<p><strong>Step 1:</strong> Since the exponent is infinite and self-similar, <span class='math'>y = (\cos x)^y</span></p><p><strong>Step 2:</strong> Take logarithm: <span class='math'>\log y = y \log(\cos x)</span></p><p><strong>Step 3:</strong> Differentiate: <span class='math'>\frac{1}{y}\frac{dy}{dx} = \frac{dy}{dx}\log(\cos x) + y \cdot \frac{-\sin x}{\cos x}</span></p><p><strong>Step 4:</strong> Solve for <span class='math'>\frac{dy}{dx} = \frac{y^2\tan x}{1-y\log(\cos x)}</span></p>
Correct Answer: B