Sets, Relations & Functions
Inverse Functions
Grade 11
Question:
<p>If <i>g</i> is the inverse of a function <i>f</i> and \(f'(x) = \dfrac{1}{1+x^5}\), then \(g'(x)\) is equal to</p>
<p>\(\dfrac{1}{1+\{g(x)\}^5}\)</p>
<p>\(1 + \{g(x)\}^5\)</p>
<p>\(1 + x^5\)</p>
<p>\(5x^4\)</p>
Step-by-Step Solution
Key Concept: If g is the inverse of f, then g'(x) = 1/f'(g(x)). You must substitute g(x) into the derivative of f, not evaluate f' at x.
<p><strong>Step 1:</strong> Recall the inverse function derivative formula: If g = f⁻¹, then g'(x) = 1/f'(g(x))</p><p><strong>Step 2:</strong> We are given f'(x) = 1/(1+x⁵)</p><p><strong>Step 3:</strong> Apply the formula: g'(x) = 1/f'(g(x)) = 1/[1/(1+[g(x)]⁵)] = 1 + [g(x)]⁵</p><p><strong>Step 4:</strong> Since g(x) is the inverse of f, we have f(g(x)) = x, so [g(x)]⁵ can be expressed in terms of the relationship, but the direct form is:</p><p>g'(x) = 1 + [g(x)]⁵</p><p>∴ Answer: B</p>
Correct Answer: B