Differentiate w.r.t.x: (ax +b)

Answer

To differentiate the function \( y = ax + b \) with respect to \( x \), follow these steps:

### Step 1: Identify the function
We start with the function:
\[ y = ax + b \]

### Step 2: Apply the differentiation rules
To differentiate \( y \) with respect to \( x \), we use the rule that states the derivative of a constant multiplied by a variable is the constant itself, and the derivative of a constant is zero.

### Step 3: Differentiate each term
We can separate the differentiation of the function into two parts:
\[ \frac{dy}{dx} = \frac{d}{dx}(ax) + \frac{d}{dx}(b) \]

### Step 4: Differentiate \( ax \)
Since \( a \) is a constant, we have:
\[ \frac{d}{dx}(ax) = a \cdot \frac{d}{dx}(x) = a \cdot 1 = a \]

### Step 5: Differentiate \( b \)
Since \( b \) is also a constant, we have:
\[ \frac{d}{dx}(b) = 0 \]

### Step 6: Combine the results
Now, we combine the results from the differentiation:
\[ \frac{dy}{dx} = a + 0 = a \]

### Final Answer
Thus, the derivative of \( y = ax + b \) with respect to \( x \) is:
\[ \frac{dy}{dx} = a \]

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