Limits, Continuity & Differentiability
Chain Rule and Derivatives
Grade 12
Question:
<p>Let <i>f</i> and <i>g</i> be differentiable functions satisfying <i>g</i>'(<i>a</i>) = 2, <i>g</i>(<i>a</i>) = <i>b</i> and <i>f</i>∘<i>g</i> = <i>I</i> (Identity function). Then, <i>f</i>'(<i>b</i>) is equal to</p>
<p>(a) 2</p>
<p>(b) 2/3</p>
<p>(c) 1/2</p>
<p>(d) None of these</p>
Step-by-Step Solution
Key Concept: If f∘g = I (identity), then f and g are inverse functions, so their derivatives are reciprocals: f'(b)·g'(a) = 1.
<p><strong>Step 1:</strong> Given: <i>f</i>∘<i>g</i> = <i>I</i>, which means <i>f</i>(<i>g</i>(<i>x</i>)) = <i>x</i> for all <i>x</i> ∈ ℝ</p><p><strong>Step 2:</strong> Differentiating both sides: <i>f</i>'(<i>g</i>(<i>x</i>)) · <i>g</i>'(<i>x</i>) = 1</p><p><strong>Step 3:</strong> At <i>x</i> = <i>a</i>: <i>f</i>'(<i>g</i>(<i>a</i>)) · <i>g</i>'(<i>a</i>) = 1</p><p><strong>Step 4:</strong> Since <i>g</i>(<i>a</i>) = <i>b</i> and <i>g</i>'(<i>a</i>) = 2: <i>f</i>'(<i>b</i>) · 2 = 1</p><p><strong>Step 5:</strong> Therefore, <i>f</i>'(<i>b</i>) = 1/2</p><p>∴ Answer is (c).</p>
Correct Answer: c