Applications of Derivatives
Successive Derivatives
Grade 12
Question:
<p>If <span class='math'>y = 3\cos(\log x) + 4\sin(\log x)</span>, then <span class='math'>x^2 y_2 + xy_1</span> is equal to</p>
<p>(a) <span class='math'>y</span></p>
<p>(b) <span class='math'>-xy</span></p>
<p>(c) <span class='math'>-y</span></p>
<p>(d) <span class='math'>0</span></p>
Step-by-Step Solution
Key Concept: Use successive differentiation and logarithmic differentiation properties to find the relationship.
<p><strong>Step 1:</strong> Find first derivative: <span class='math'>y_1 = -\frac{3\sin(\log x)}{x} + \frac{4\cos(\log x)}{x}</span></p><p><strong>Step 2:</strong> Find second derivative: <span class='math'>y_2 = -\frac{3\cos(\log x)}{x^2} - \frac{3\sin(\log x)}{x^2} - \frac{4\sin(\log x)}{x^2} + \frac{4\cos(\log x)}{x^2}</span></p><p><strong>Step 3:</strong> Compute <span class='math'>x^2 y_2 + xy_1 = -3\cos(\log x) - 3\sin(\log x) - 4\sin(\log x) + 4\cos(\log x) - 3\sin(\log x) + 4\cos(\log x) = -y</span></p>
Correct Answer: C