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: Note the kind of function.
We have $y = 15\cos x - 13\sin x$, a mix of sine and cosine. A handy fact is that any such combination flips sign after two differentiations.

Step 2: First derivative.
Differentiate once. Since $\cos$ gives $-\sin$ and $\sin$ gives $\cos$, \[ \frac{dy}{dx} = -15\sin x - 13\cos x \]

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

Step 4: Match it back to $y$.
Factor out a minus sign: $-15\cos x + 13\sin x = -(15\cos x - 13\sin x) = -y$.

Step 5: Answer.
So the second derivative is just the negative of the original function. \[ \boxed{\dfrac{d^2y}{dx^2} = -y} \]