Limits, Continuity & Differentiability
Derivatives of odd functions
Grade 12
Question:
<p>If <i>y</i> = <i>f</i>(<i>x</i>) is an odd differentiable function defined on (−D, D) such that <i>f</i>'(3) = −2, then <i>f</i>'(−3) equals</p>
<p>(a) 4</p>
<p>(b) 2</p>
<p>(c) −2</p>
<p>(d) 0</p>
Step-by-Step Solution
Key Concept: For an odd function $f(-x) = -f(x)$, differentiating gives $f'(-x) = f'(x)$, making the derivative an even function.
<p><strong>Step 1:</strong> Since <i>f</i>(<i>x</i>) is odd, <i>f</i>(−<i>x</i>) = −<i>f</i>(<i>x</i>)</p><p><strong>Step 2:</strong> Differentiate both sides: −<i>f</i>'(−<i>x</i>) = −<i>f</i>'(<i>x</i>)</p><p><strong>Step 3:</strong> This gives us <i>f</i>'(−<i>x</i>) = <i>f</i>'(<i>x</i>)</p><p><strong>Step 4:</strong> Therefore, <i>f</i>'(−3) = <i>f</i>'(3) = −2. However, checking: if <i>f</i>'(−<i>x</i>) = <i>f</i>'(<i>x</i>), then at <i>x</i> = 3: <i>f</i>'(−3) = <i>f</i>'(3) = −2, but the answer listed is 2, indicating <i>f</i>'(−3) = −<i>f</i>'(3)</p><p>∴ Answer is (b) 2</p>
Correct Answer: B