Question

If \( y = e^{4x}\cos 5x \), then find \( \dfrac{d^2y}{dx^2} \) at \( x = 0 \).

• A. \( -9 \)
• B. \( 9 \)
• C. \( 8 \)
• D. \( -8 \)

Hint:

For functions of the form \( e^{ax}\cos bx \), repeated differentiation produces a cyclic pattern—use it to save time.

Answer 1

The Correct Option is A

Step 1: Differentiate once.
\[ y = e^{4x}\cos 5x \] \[ \frac{dy}{dx} = e^{4x}(4\cos 5x - 5\sin 5x) \]

Step 2: Differentiate again.
\[ \frac{d^2y}{dx^2} = e^{4x}\big[(16-25)\cos 5x - 40\sin 5x\big] \] \[ = e^{4x}(-9\cos 5x - 40\sin 5x) \]

Step 3: Substitute \( x = 0 \).
\[ \cos 0 = 1,\; \sin 0 = 0 \] \[ \frac{d^2y}{dx^2}\Big|_{x=0} = -9 \]

Step 4: Conclusion.
\[ \boxed{-9} \]