Question

If \( y(x)=15\cos(x)-13\sin(x) \), then \( \dfrac{d^2y}{dx^2} \) will be

• A. \(2\)
• B. \(\dfrac{\pi}{y}\)
• C. \(-y\)
• D. \(\dfrac{y^2}{x}\)

Hint:

For functions involving \(\sin x\) and \(\cos x\), the second derivative often becomes the negative of the original function.

Answer 1

The Correct Option is C

Step 1: Write the given function.
\[ y(x)=15\cos x-13\sin x \]

Step 2: Differentiate once with respect to \(x\).
Using
\[ \frac{d}{dx}(\cos x)=-\sin x \] and
\[ \frac{d}{dx}(\sin x)=\cos x \] we get
\[ \frac{dy}{dx} = -15\sin x-13\cos x \]

Step 3: Differentiate again to find second derivative.
\[ \frac{d^2y}{dx^2} = -15\cos x+13\sin x \]

Step 4: Compare with the original function.
Original function is
\[ y=15\cos x-13\sin x \]
Taking negative of \(y\),
\[ -y=-15\cos x+13\sin x \]

Step 5: Observe the equality.
\[ \frac{d^2y}{dx^2}=-y \]

Step 6: Verify the expression carefully.
Both expressions are identical term by term. Hence, the second derivative is equal to \(-y\).

Step 7: Final conclusion.
\[ \boxed{\frac{d^2y}{dx^2}=-y} \]
Therefore, the correct answer is option (C).