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><th>\(x\)</th><td>0</td><td>1</td><td>2</td><td>3</td><td>4</td></tr><tr><th>\(f(x)\)</th><td>\(-2\)</td><td>\(-1\)</td><td>2</td><td>4</td><td>6</td></tr><tr><th>\(f'(x)\)</th><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 chain rule on g(x) = 1/f⁻¹(x) to get g'(x) = -f⁻¹'(x)/[f⁻¹(x)]², then apply the inverse function derivative formula: (f⁻¹)'(y) = 1/f'(f⁻¹(y)).
<p><strong>Step 1:</strong> Identify what we need. We have g(x) = 1/f⁻¹(x), so g'(4) = -[f⁻¹'(4)]/[f⁻¹(4)]²</p><p><strong>Step 2:</strong> Find f⁻¹(4) from the table. Since f(3) = 4, we have f⁻¹(4) = 3.</p><p><strong>Step 3:</strong> Find f⁻¹'(4) using the inverse derivative formula: (f⁻¹)'(4) = 1/f'(f⁻¹(4)) = 1/f'(3) = 1/(4/3) = 3/4</p><p><strong>Step 4:</strong> Calculate g'(4) = -[f⁻¹'(4)]/[f⁻¹(4)]² = -(3/4)/(3)² = -(3/4)/9 = -3/36 = -1/12</p><p>∴ Answer: A</p>
Correct Answer: A