Use delta method to find the derivatives of the following :
`sinx+cosx`

Answer

To find the derivative of the function \( y = \sin x + \cos x \) using the delta method, we follow these steps:

### Step 1: Define the function and increment
Let \( y = \sin x + \cos x \). We consider a small increment \( \Delta x \) in \( x \), leading to a new value \( x + \Delta x \). Therefore, the new value of \( y \) becomes:
\[ y + \Delta y = \sin(x + \Delta x) + \cos(x + \Delta x) \]

### Step 2: Express \( \Delta y \)
Now, we express \( \Delta y \):
\[ \Delta y = \sin(x + \Delta x) + \cos(x + \Delta x) - (\sin x + \cos x) \]

### Step 3: Apply trigonometric identities
Using the angle addition formulas:
\[ \sin(x + \Delta x) = \sin x \cos(\Delta x) + \cos x \sin(\Delta x) \]
\[ \cos(x + \Delta x) = \cos x \cos(\Delta x) - \sin x \sin(\Delta x) \]
Substituting these into the expression for \( \Delta y \):
\[ \Delta y = [\sin x \cos(\Delta x) + \cos x \sin(\Delta x)] + [\cos x \cos(\Delta x) - \sin x \sin(\Delta x)] - (\sin x + \cos x) \]

### Step 4: Simplify \( \Delta y \)
Now, we simplify \( \Delta y \):
\[ \Delta y = \sin x \cos(\Delta x) + \cos x \sin(\Delta x) + \cos x \cos(\Delta x) - \sin x \sin(\Delta x) - \sin x - \cos x \]
Grouping the terms:
\[ \Delta y = \sin x (\cos(\Delta x) - 1) + \cos x (\cos(\Delta x) - 1) + \cos x \sin(\Delta x) - \sin x \sin(\Delta x) \]

### Step 5: Factor out common terms
We can factor out \( \cos(\Delta x) - 1 \) and \( \sin(\Delta x) \):
\[ \Delta y = (\sin x + \cos x)(\cos(\Delta x) - 1) + (\cos x - \sin x) \sin(\Delta x) \]

### Step 6: Divide by \( \Delta x \)
Next, we divide \( \Delta y \) by \( \Delta x \):
\[ \frac{\Delta y}{\Delta x} = \frac{(\sin x + \cos x)(\cos(\Delta x) - 1)}{\Delta x} + \frac{(\cos x - \sin x) \sin(\Delta x)}{\Delta x} \]

### Step 7: Take the limit as \( \Delta x \to 0 \)
Now we take the limit as \( \Delta x \to 0 \):
\[ \frac{dy}{dx} = \lim_{\Delta x \to 0} \left( \frac{(\sin x + \cos x)(\cos(\Delta x) - 1)}{\Delta x} + \frac{(\cos x - \sin x) \sin(\Delta x)}{\Delta x} \right) \]

Using the limits:
- \( \lim_{\Delta x \to 0} \frac{\sin(\Delta x)}{\Delta x} = 1 \)
- \( \lim_{\Delta x \to 0} \frac{1 - \cos(\Delta x)}{\Delta x} = 0 \)

We find:
\[ \frac{dy}{dx} = 0 + (\cos x - \sin x) \cdot 1 \]

### Final Result
Thus, the derivative of \( y = \sin x + \cos x \) is:
\[ \frac{dy}{dx} = \cos x - \sin x \]