Limits, Continuity & Differentiability
Inverse function derivative
Grade 12
Question:
<p>Let \(f\) and \(g\) be inverse functions of each other. If \(f(1) = 3\) and \(f'(1) = 4\), then \(g'(3)\) equals:</p>
<p>\(\dfrac{1}{4}\)</p>
<p>\(\dfrac{1}{3}\)</p>
<p>\(4\)</p>
<p>\(3\)</p>
Step-by-Step Solution
Key Concept: If f and g are inverse functions, then g(f(x)) = x for all x in the domain of f. Differentiating this identity using the chain rule gives the relationship g'(f(x)) · f'(x) = 1, which directly yields g'(f(a)) = 1/f'(a).
<p><strong>Step 1:</strong> Since f and g are inverse functions, we have the fundamental property: g(f(x)) = x for all x in the domain of f.</p><p><strong>Step 2:</strong> Differentiate both sides with respect to x using the chain rule:</p><p>d/dx[g(f(x))] = d/dx[x]</p><p>g'(f(x)) · f'(x) = 1</p><p><strong>Step 3:</strong> Rearrange to get the inverse derivative formula:</p><p>g'(f(x)) = 1/f'(x)</p><p><strong>Step 4:</strong> We are given that f(1) = 3 and f'(1) = 4. Substitute x = 1 into the formula from Step 3:</p><p>g'(f(1)) = 1/f'(1)</p><p>g'(3) = 1/4</p><p><strong>∴ Answer: A (g'(3) = 1/4)</strong></p>
Correct Answer: A