Evaluate differentiation of y with respect to x :
`y=5x+cosx`

Explanation

To evaluate the differentiation of \( y \) with respect to \( x \) for the function \( y = 5x + \cos x \), we will follow these steps:

### Step 1: Identify the terms in the function
The function \( y \) consists of two terms:
1. An algebraic term: \( 5x \)
2. A trigonometric term: \( \cos x \)

### Step 2: Differentiate the algebraic term
For the algebraic term \( 5x \):
- The differentiation rule states that if \( y = ax^n \), then \( \frac{dy}{dx} = nax^{n-1} \).
- Here, \( a = 5 \) and \( n = 1 \).
- Therefore, the derivative of \( 5x \) is:
\[ \frac{d}{dx}(5x) = 5 \cdot 1 \cdot x^{1-1} = 5 \]

### Step 3: Differentiate the trigonometric term
For the trigonometric term \( \cos x \):
- The differentiation rule for cosine states that \( \frac{d}{dx}(\cos x) = -\sin x \).
- Therefore, the derivative of \( \cos x \) is:
\[ \frac{d}{dx}(\cos x) = -\sin x \]

### Step 4: Combine the derivatives
Now, we combine the derivatives of both terms to find \( \frac{dy}{dx} \):
\[ \frac{dy}{dx} = \frac{d}{dx}(5x) + \frac{d}{dx}(\cos x) = 5 - \sin x \]

### Final Answer
Thus, the differentiation of \( y \) with respect to \( x \) is:
\[ \frac{dy}{dx} = 5 - \sin x \]

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