Find `(d^(2)y)/(dx^(2))" when "y=e^(x)+cosx`

Answer

To find the second derivative of the function \( y = e^x + \cos x \), we will follow these steps:

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

Given:
\[ y = e^x + \cos x \]

We differentiate \( y \) with respect to \( x \):
\[ \frac{dy}{dx} = \frac{d}{dx}(e^x) + \frac{d}{dx}(\cos x) \]

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

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

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

Now, we differentiate \( \frac{dy}{dx} \) to find \( \frac{d^2y}{dx^2} \):
\[ \frac{d^2y}{dx^2} = \frac{d}{dx}(e^x - \sin x) \]

Again, using the known derivatives:
- The derivative of \( e^x \) is \( e^x \).
- The derivative of \( -\sin x \) is \( -\cos x \).

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

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

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