Limits, Continuity & Differentiability
Derivatives of Inverse Functions
Grade 12
Question:
<p>Let <span>f(x)</span> be the inverse of the function <span>g(x)</span> and <span>f'(x) = 1/(5·g(x)^5)</span>, then <span>d/dx[f(x)]</span> is equal to</p>
<p>(a) <span>1/(1+[f(x)]^5)</span></p>
<p>(b) <span>1/(1+[f(x)]^5)</span></p>
<p>(c) <span>1+[f(x)]^5</span></p>
<p>(d) <span>1+f(x)</span></p>
Step-by-Step Solution
Key Concept: The derivative of an inverse function satisfies f'(x) = 1/g'(f(x)); use the given condition to find the relationship.
<p>If <span>f</span> is the inverse of <span>g</span>, then <span>f(g(x)) = x</span>.</p><p>Differentiating: <span>f'(g(x)) · g'(x) = 1</span></p><p>So <span>f'(g(x)) = 1/g'(x)</span></p><p>Given <span>f'(x) = 1/(5·[g(x)]^5)</span>, and using the inverse function property:</p><p><span>f'(f(x)) = 1/g'(f(x))</span></p><p>This yields <span>f'(x) = 1/(1+[f(x)]^5)</span></p>
Correct Answer: A