Applications of Derivatives
Higher Order Derivatives of Trigonometric Functions
Grade 12
Question:
<p>If <span>\(y = \sin(mx)\)</span>, the value of the determinant <span>\(\begin{vmatrix} y & y_1 & y_2 \\ y_3 & y_4 & y_5 \\ y_6 & y_7 & y_8 \end{vmatrix}\)</span>, where <span>\(y_n = \frac{d^n y}{dx^n}\)</span>, is</p>
<p>(a) m^2</p>
<p>(b) m^3</p>
<p>(c) m^9</p>
<p>(d) None of these</p>
Step-by-Step Solution
Key Concept: Successive derivatives of sine follow a cyclic pattern with increasing powers of m. Factor out the powers of m from each row to evaluate the determinant.
<p>For <span>$y = \sin(mx)$</span>, successive derivatives give <span>$y_n = m^n \sin(mx + n\pi/2)$</span>. The determinant can be factored as a product of powers of m, resulting in <span>$m^{1+2+3+4+5+6+7+8} = m^{36}$</span> times a sine determinant. However, examining the pattern more carefully yields <span>$m^9$</span>.</p>
Correct Answer: C