If `y=sinx+cosx" then "(d^(2)y)/(dx^(2))` is :-

Options

A. `sinx-cosx`
B. `cosx-sinx`
C. `-(sinx+cosx)`
D. None of these

Answer

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

### Step 1: Differentiate \( y \) with respect to \( x \)

Given:
\[ y = \sin x + \cos x \]

To find the first derivative \( \frac{dy}{dx} \), we differentiate \( y \):
\[ \frac{dy}{dx} = \frac{d}{dx}(\sin x) + \frac{d}{dx}(\cos x) \]

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

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

### Step 2: Differentiate \( \frac{dy}{dx} \) to find \( \frac{d^2y}{dx^2} \)

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

Differentiating each term:
- The derivative of \( \cos x \) is \( -\sin x \).
- The derivative of \( -\sin x \) is \( -\cos x \).

Putting it together:
\[ \frac{d^2y}{dx^2} = -\sin x - \cos x \]

### Final Answer

Thus, the second derivative \( \frac{d^2y}{dx^2} \) is:
\[ \frac{d^2y}{dx^2} = -\sin x - \cos x \]