Relations & Functions
Inverse Functions and Differentiation
Grade 12
Question:
<p>Given <em>g</em>(<em>x</em>) = <em>f</em><sup>−1</sup>(<em>x</em>). Therefore, <em>f</em>(<em>g</em>(<em>x</em>)) = <em>x</em>. If <em>f</em>(<em>x</em>) = <em>x</em> + {<em>x</em>}<sup>5</sup>, then <em>g</em>′(<em>x</em>) equals:</p>
<p>1 + {<em>g</em>(<em>x</em>)}<sup>5</sup></p>
<p>\(\dfrac{1}{1+\{g(x)\}^5}\)</p>
<p>1 − {<em>g</em>(<em>x</em>)}<sup>5</sup></p>
<p>\(\dfrac{1}{1-\{g(x)\}^5}\)</p>
Step-by-Step Solution
Key Concept: Since g(x) = f⁻¹(x), differentiating f(g(x)) = x implicitly gives f'(g(x))·g'(x) = 1, so g'(x) = 1/f'(g(x)). You must find f'(x) first, then express it in terms of g(x).
<p><strong>Step 1:</strong> Find f'(x). Given f(x) = x + {x}/5, where {x} is the fractional part of x.</p><p>f'(x) = 1 + (1/5)·d/dx[{x}] = 1 + 1/5 = 6/5 (where {x} is differentiable)</p><p><strong>Step 2:</strong> Use the inverse function derivative formula. Since g(x) = f⁻¹(x), differentiating f(g(x)) = x:</p><p>f'(g(x))·g'(x) = 1</p><p><strong>Step 3:</strong> Substitute f'(g(x)) = 6/5:</p><p>(6/5)·g'(x) = 1</p><p>g'(x) = 5/6</p><p>∴ Answer: B</p>
Correct Answer: B