Limits, Continuity & Differentiability
Properties of derivatives
Grade 12
Question:
<p>Let <i>f</i>(<i>x</i>) = <i>x</i><sup><i>n</i></sup>, <i>n</i> being a non-negative integer. The value of <i>n</i> for which the equality <i>f</i>'(<i>x</i> + <i>y</i>) = <i>f</i>'(<i>x</i>) + <i>f</i>'(<i>y</i>) is valid for all <i>x</i>, <i>y</i> ≠ 0, is</p>
<p>(a) 0, 1</p>
<p>(b) 1, 2</p>
<p>(c) 2, 4</p>
<p>(d) None of these</p>
Step-by-Step Solution
Key Concept: The additive property of derivatives $f'(x+y) = f'(x) + f'(y)$ only holds for linear functions (when the exponent is 1) and the zero function.
<p><strong>Step 1:</strong> For <i>f</i>(<i>x</i>) = <i>x</i><sup><i>n</i></sup>, we have <i>f</i>'(<i>x</i>) = <i>nx</i><sup><i>n</i>−1</sup></p><p><strong>Step 2:</strong> Check <i>f</i>'(<i>x</i> + <i>y</i>) = <i>n</i>(<i>x</i> + <i>y</i>)<sup><i>n</i>−1</sup> and <i>f</i>'(<i>x</i>) + <i>f</i>'(<i>y</i>) = <i>nx</i><sup><i>n</i>−1</sup> + <i>ny</i><sup><i>n</i>−1</sup></p><p><strong>Step 3:</strong> For equality to hold for all <i>x</i>, <i>y</i>, we need <i>n</i>(<i>x</i> + <i>y</i>)<sup><i>n</i>−1</sup> = <i>nx</i><sup><i>n</i>−1</sup> + <i>ny</i><sup><i>n</i>−1</sup></p><p><strong>Step 4:</strong> This is true only when <i>n</i> = 0 (making both sides 0) or <i>n</i> = 1 (both sides equal <i>n</i>)</p><p>∴ Answer is (a) 0, 1</p>
Correct Answer: A