Applications of Derivatives
Successive Derivatives
Grade 12
Question:
<p>If <span class='math'>y = 500e^{7x} + 600e^{-7x}</span>, then the value of <span class='math'>\frac{d^2y}{dx^2}</span> is</p>
<p>(a) <span class='math'>y</span></p>
<p>(b) <span class='math'>7y</span></p>
<p>(c) <span class='math'>40y</span></p>
<p>(d) <span class='math'>49y</span></p>
Step-by-Step Solution
Key Concept: Differentiate exponential functions and recognize the pattern in successive derivatives.
<p><strong>Step 1:</strong> Find first derivative: <span class='math'>\frac{dy}{dx} = 500 \cdot 7e^{7x} - 600 \cdot 7e^{-7x} = 3500e^{7x} - 4200e^{-7x}</span></p><p><strong>Step 2:</strong> Find second derivative: <span class='math'>\frac{d^2y}{dx^2} = 3500 \cdot 7e^{7x} + 4200 \cdot 7e^{-7x} = 49(500e^{7x} + 600e^{-7x}) = 49y</span></p>
Correct Answer: D