If y = 5 cos x - 3 sin x, then `(d^2y)/(dx^2)` is equal to :

Options

A. `-y`
B. y
C. 25y
D. 9y

Answer

To solve the problem, we need to find the second derivative of the function \( y = 5 \cos x - 3 \sin x \). Let's go through the steps systematically.

### Step 1: Find the first derivative \( \frac{dy}{dx} \)

Given:
\[ y = 5 \cos x - 3 \sin x \]

To find the first derivative, we differentiate \( y \) with respect to \( x \):
\[ \frac{dy}{dx} = \frac{d}{dx}(5 \cos x) - \frac{d}{dx}(3 \sin x) \]

Using the derivatives of trigonometric functions:
- The derivative of \( \cos x \) is \( -\sin x \)
- The derivative of \( \sin x \) is \( \cos x \)

Thus, we have:
\[ \frac{dy}{dx} = 5(-\sin x) - 3(\cos x) \]
\[ \frac{dy}{dx} = -5 \sin x - 3 \cos x \]

### Step 2: Find the second derivative \( \frac{d^2y}{dx^2} \)

Now, we differentiate \( \frac{dy}{dx} \) to find the second derivative:
\[ \frac{d^2y}{dx^2} = \frac{d}{dx}(-5 \sin x - 3 \cos x) \]

Again, using the derivatives of trigonometric functions:
\[ \frac{d^2y}{dx^2} = -5 \frac{d}{dx}(\sin x) - 3 \frac{d}{dx}(\cos x) \]
\[ \frac{d^2y}{dx^2} = -5(\cos x) - 3(-\sin x) \]
\[ \frac{d^2y}{dx^2} = -5 \cos x + 3 \sin x \]

### Step 3: Express \( \frac{d^2y}{dx^2} \) in terms of \( y \)

Recall that:
\[ y = 5 \cos x - 3 \sin x \]

We can rearrange this equation:
\[ 5 \cos x - 3 \sin x = y \]

Now, we can express \( -5 \cos x + 3 \sin x \) in terms of \( y \):
\[ -5 \cos x + 3 \sin x = - (5 \cos x - 3 \sin x) = -y \]

Thus, we have:
\[ \frac{d^2y}{dx^2} = -y \]

### Final Answer
\[ \frac{d^2y}{dx^2} = -y \]