Limits, Continuity & Differentiability
Differentiation of Products
Grade 12
Question:
<p>If <span>f(x) = cos x × cos 2x × cos 4x × cos 8x × cos 16x</span>, then <span>f'(π/4)</span> is</p>
<p>(a) 1</p>
<p>(b) 2</p>
<p>(c) 1/2</p>
<p>(d) 0</p>
Step-by-Step Solution
Key Concept: When a product includes a factor that equals zero at a point, the derivative can be found by recognizing that f(π/4)=0 due to cos(2·π/4)=cos(π/2)=0.
<p>To find <span>f'(x)</span> for the product <span>f(x) = cos x × cos 2x × cos 4x × cos 8x × cos 16x</span>, use the product rule.</p><p>Using logarithmic differentiation:</p><p><span>ln f(x) = ln(cos x) + ln(cos 2x) + ln(cos 4x) + ln(cos 8x) + ln(cos 16x)</span></p><p>Differentiating both sides:</p><p><span>f'(x)/f(x) = -tan x - 2tan 2x - 4tan 4x - 8tan 8x - 16tan 16x</span></p><p>At <span>x = π/4</span>: <span>cos(π/4) = 1/√2</span>, <span>cos(π/2) = 0</span></p><p>Since <span>cos 2x = cos(π/2) = 0</span> when <span>x = π/4</span>, we have <span>f(π/4) = 0</span></p><p>Therefore <span>f'(π/4) = 0</span></p>
Correct Answer: D