<p>If \(f(x) = x^4 \tan x^3 - x \ln(1 + x^2)\), then the value of \(\dfrac{d^4 f(x)}{dx^4}\) at \(x = 0\) is:</p>
Step-by-Step Solution
Key Concept: Use Taylor series expansion of each term around x=0 to find the coefficient of x⁴, then multiply by 4! to get the fourth derivative at x=0. The fourth derivative at a point equals 4! times the coefficient of x⁴ in the Taylor series.
<p><strong>Step 1:</strong> Expand each term using Taylor series.</p><p>For <strong>x⁴tan(x³)</strong>: tan(u) = u + u³/3 + ..., so tan(x³) = x³ + x⁹/3 + ...<br>Therefore: x⁴tan(x³) = x⁴(x³ + x⁹/3 + ...) = x⁷ + x¹³/3 + ...<br>The x⁴ coefficient is <strong>0</strong>.</p><p><strong>Step 2:</strong> Expand <strong>x·ln(1+x²)</strong>: ln(1+u) = u - u²/2 + u³/3 - ..., so ln(1+x²) = x² - x⁴/2 + x⁶/3 - ...<br>Therefore: x·ln(1+x²) = x(x² - x⁴/2 + ...) = x³ - x⁵/2 + ...<br>The x⁴ coefficient is <strong>0</strong>.</p><p><strong>Step 3:</strong> Combine: f(x) = x⁷ + ... - (x³ - x⁵/2 + ...)<br>The coefficient of x⁴ in f(x) is <strong>0</strong>.</p><p><strong>Step 4:</strong> Apply the relation: f⁴(0) = 4! × (coefficient of x⁴) = 24 × 0 = <strong>0</strong>.</p><p>∴ Answer: A (which is <strong>0</strong>)</p>
Correct Answer: A