Applications of Derivatives
Higher Order Derivatives
Grade 12
Question:
<p>Let <span>\(f(x) = \begin{vmatrix} x^3 + \sin x & \cos x & 1 \\ p^2 & p & 0 \\ p^3 & p^2 & p \end{vmatrix}\)</span>, where p is a constant. Then <span>\(\frac{d^{33}}{dx^{33}}\{f(x)\}\)</span> at <span>\(x = 0\)</span> is</p>
<p>(a) p</p>
<p>(b) p + p^2</p>
<p>(c) p + p^3</p>
<p>(d) independent of p</p>
Step-by-Step Solution
Key Concept: The determinant contains a constant matrix (rows 2 and 3) multiplied by varying terms in row 1. High-order derivatives of bounded trigonometric functions at zero follow a pattern.
<p>Expand the determinant and identify the terms. The 33rd derivative at x=0 will depend on the nature of trigonometric and polynomial terms. Since the second and third rows are independent of x, only the first row contributes. The sine and cosine terms will vanish after multiple differentiation at x=0, showing independence from p.</p>
Correct Answer: D