Limits, Continuity & Differentiability
Differentiability
Grade 12
Question:
<p>Let \( f: (-1, 1) \to R \) be a differentiable function with \( f(0) = -1 \) and \( f'(0) = 1 \). Let \( g(x) = [f(2f(x)+2)]^2 \). Then \( g'(0) = \)</p>
<p>\(-4\)</p>
<p>\(0\)</p>
<p>\(-2\)</p>
<p>\(4\)</p>
Step-by-Step Solution
Key Concept: Use the chain rule systematically: g'(x) requires differentiating the outer square, then the composition f(2f(x)+2), then the inner 2f(x)+2. Crucially, evaluate f and f' at the correct points using given values f(0)=-1 and f'(0)=1.
<p><strong>Step 1:</strong> Find g'(x) using the chain rule.</p><p>g(x) = [f(2f(x)+2)]²</p><p>g'(x) = 2·f(2f(x)+2)·f'(2f(x)+2)·d/dx[2f(x)+2]</p><p><strong>Step 2:</strong> Simplify the derivative of the innermost function.</p><p>d/dx[2f(x)+2] = 2f'(x)</p><p>So: g'(x) = 2·f(2f(x)+2)·f'(2f(x)+2)·2f'(x) = 4f'(x)·f(2f(x)+2)·f'(2f(x)+2)</p><p><strong>Step 3:</strong> Evaluate at x = 0.</p><p>• f(0) = -1 (given)</p><p>• f'(0) = 1 (given)</p><p>• 2f(0) + 2 = 2(-1) + 2 = 0</p><p>• f(2f(0)+2) = f(0) = -1</p><p>• f'(2f(0)+2) = f'(0) = 1</p><p><strong>Step 4:</strong> Substitute into g'(0).</p><p>g'(0) = 4·f'(0)·f(0)·f'(0) = 4·(1)·(-1)·(1) = -4</p><p>∴ Answer: A (g'(0) = -4)</p>
Correct Answer: A