Limits, Continuity & Differentiability
Differentiability of inverse functions
Grade 12
Question:
<p>If \(f(x) = x^2 - x + 5\), \(x > \dfrac{1}{2}\) and \(g(x)\) is its inverse function, then \(g'(7)\) equals</p>
<p>\(-\dfrac{1}{3}\)</p>
<p>\(\dfrac{1}{13}\)</p>
<p>\(\dfrac{1}{3}\)</p>
<p>\(-\dfrac{1}{13}\)</p>
Step-by-Step Solution
Key Concept: To find g'(a) where g is the inverse of f, use the inverse function derivative formula: g'(a) = 1/f'(g(a)). First find g(7) by solving f(x) = 7, then apply the formula.
<p><strong>Step 1:</strong> Find g(7) by solving f(x) = 7.</p><p>x² - x + 5 = 7</p><p>x² - x - 2 = 0</p><p>(x - 2)(x + 1) = 0</p><p>Since x > 1/2, we have g(7) = 2</p><p><strong>Step 2:</strong> Use the inverse function derivative formula: g'(a) = 1/f'(g(a))</p><p>f'(x) = 2x - 1</p><p>f'(g(7)) = f'(2) = 2(2) - 1 = 3</p><p><strong>Step 3:</strong> Therefore, g'(7) = 1/f'(g(7)) = 1/3</p><p>∴ Answer: g'(7) = <strong>1/3</strong></p>
Correct Answer: B