Limits, Continuity & Differentiability
Differentiation of inverse functions
Grade 12
Question:
<p><strong>197.</strong> Let \(g(x) = \dfrac{1}{f^{-1}(x)}\). Given the following data:</p><table border='1' cellpadding='4'><tr><td>\(x\)</td><td>0</td><td>1</td><td>2</td><td>3</td><td>4</td></tr><tr><td>\(f(x)\)</td><td>\(-2\)</td><td>\(-1\)</td><td>2</td><td>4</td><td>6</td></tr><tr><td>\(f'(x)\)</td><td>1/2</td><td>2/3</td><td>1</td><td>4/3</td><td>5/3</td></tr></table><p>The value of \(g'(4)\) is:</p>
<p>(a) \(\dfrac{-1}{12}\)</p>
<p>(b) \(\dfrac{-1}{15}\)</p>
<p>(c) \(\dfrac{1}{12}\)</p>
<p>(d) \(\dfrac{1}{15}\)</p>
Step-by-Step Solution
Key Concept: Use the derivative of inverse function formula: (f⁻¹)'(x) = 1/f'(f⁻¹(x)), then apply quotient rule to g(x) = 1/(f⁻¹(x)). The critical step is identifying that f⁻¹(4) = 2 from the table (since f(2) = 4).
<p><strong>Step 1:</strong> Find f⁻¹(4) from the table. Since f(2) = 4, we have f⁻¹(4) = 2.</p><p><strong>Step 2:</strong> Differentiate g(x) = 1/(f⁻¹(x)) using the chain rule:</p><p>g'(x) = -1/[f⁻¹(x)]² · (f⁻¹)'(x)</p><p><strong>Step 3:</strong> Use the inverse function derivative formula: (f⁻¹)'(x) = 1/f'(f⁻¹(x))</p><p>So (f⁻¹)'(4) = 1/f'(f⁻¹(4)) = 1/f'(2) = 1/(2/3) = 3/2</p><p><strong>Step 4:</strong> Substitute into the derivative formula:</p><p>g'(4) = -1/[f⁻¹(4)]² · (f⁻¹)'(4) = -1/(2)² · (3/2) = -1/4 · 3/2 = <strong>-3/8</strong></p><p>∴ Answer: A</p>
Correct Answer: A