Limits, Continuity & Differentiability
Properties of derivatives
Grade 12
<p>The derivative of an even function is an odd function.</p><p><strong>State whether the statement is true or false.</strong></p>
Step-by-Step Solution
Key Concept: If f(x) is even, then f(-x) = f(x). Differentiating both sides with respect to x using the chain rule: f'(-x)·(-1) = f'(x), which gives f'(-x) = -f'(x), proving f'(x) is odd.
<p><strong>Step 1:</strong> Let f(x) be an even function, so f(-x) = f(x) for all x in the domain.</p><p><strong>Step 2:</strong> Differentiate both sides with respect to x: d/dx[f(-x)] = d/dx[f(x)]</p><p><strong>Step 3:</strong> Apply chain rule on the left side: f'(-x)·(-1) = f'(x)</p><p><strong>Step 4:</strong> Simplify: f'(-x) = -f'(x)</p><p><strong>Step 5:</strong> This is the definition of an odd function. Therefore, the derivative of an even function is always an odd function.</p><p><strong>Verification:</strong> Example: f(x) = x² (even) → f'(x) = 2x (odd) ✓</p><p>∴ <strong>Answer: TRUE (Statement is correct)</strong></p>
Correct Answer: A