Limits, Continuity & Differentiability
Differentiation of Inverse Functions
Grade 12
Question:
<p>Let \(f: R \to R\) defined by \(f(x) = x^3 + 3x + 1\) and \(g\) be the inverse of \(f\), then the value of \(g''(5)\) equals:</p>
<p>(a) \(\dfrac{1}{6}\)</p>
<p>(b) \(\dfrac{-1}{6}\)</p>
<p>(c) \(\dfrac{1}{36}\)</p>
<p>(d) \(\dfrac{-1}{36}\)</p>
Step-by-Step Solution
Key Concept: Use implicit differentiation on the inverse function relationship f(g(x)) = x to find derivatives of g, then apply the formula g''(x) = -f''(g(x))/[f'(g(x))]³.
<p><strong>Step 1:</strong> Find g(5) by solving f(x) = 5.</p><p>x³ + 3x + 1 = 5 ⟹ x³ + 3x - 4 = 0</p><p>Testing x = 1: 1 + 3 - 4 = 0 ✓</p><p>So g(5) = 1</p><p><strong>Step 2:</strong> Find f'(x) and evaluate at x = 1.</p><p>f'(x) = 3x² + 3</p><p>f'(1) = 3(1)² + 3 = 6</p><p><strong>Step 3:</strong> Find g'(5) using inverse derivative formula.</p><p>g'(5) = 1/f'(g(5)) = 1/f'(1) = 1/6</p><p><strong>Step 4:</strong> Find f''(x) and evaluate at x = 1.</p><p>f''(x) = 6x</p><p>f''(1) = 6</p><p><strong>Step 5:</strong> Use the formula for second derivative of inverse function.</p><p>g''(x) = -f''(g(x))/[f'(g(x))]³</p><p>g''(5) = -f''(g(5))/[f'(g(5))]³ = -f''(1)/[f'(1)]³ = -6/6³ = -6/216 = -1/36</p><p>∴ Answer: D</p>
Correct Answer: D