Find the derivatives of the following :
`5secx+4cosx`

Answer

To find the derivative of the function \( y = 5 \sec x + 4 \cos x \), we will differentiate each term separately using the rules of differentiation.

### Step-by-Step Solution:

1. **Identify the function**:
\[ y = 5 \sec x + 4 \cos x \]

2. **Differentiate each term**:
We will use the following differentiation rules:
- The derivative of \( \sec x \) is \( \sec x \tan x \).
- The derivative of \( \cos x \) is \( -\sin x \).

Therefore, we can differentiate \( y \):
\[ \frac{dy}{dx} = \frac{d}{dx}(5 \sec x) + \frac{d}{dx}(4 \cos x) \]

3. **Differentiate \( 5 \sec x \)**:
\[ \frac{d}{dx}(5 \sec x) = 5 \cdot \frac{d}{dx}(\sec x) = 5 \sec x \tan x \]

4. **Differentiate \( 4 \cos x \)**:
\[ \frac{d}{dx}(4 \cos x) = 4 \cdot \frac{d}{dx}(\cos x) = 4 \cdot (-\sin x) = -4 \sin x \]

5. **Combine the derivatives**:
Now we combine the results from steps 3 and 4:
\[ \frac{dy}{dx} = 5 \sec x \tan x - 4 \sin x \]

### Final Answer:
\[ \frac{dy}{dx} = 5 \sec x \tan x - 4 \sin x \]