`sqrt(ax +b)`

Answer

To differentiate the function \( \sqrt{ax + b} \) with respect to \( x \), we can follow these steps:

### Step 1: Rewrite the Function
We start by rewriting the square root in exponent form:
\[ y = \sqrt{ax + b} = (ax + b)^{\frac{1}{2}} \]

### Step 2: Apply the Chain Rule
To differentiate \( y \) with respect to \( x \), we will use the chain rule. The chain rule states that if you have a composite function \( f(g(x)) \), then the derivative is:
\[ \frac{dy}{dx} = f'(g(x)) \cdot g'(x) \]
In our case, let \( f(u) = u^{\frac{1}{2}} \) where \( u = ax + b \).

### Step 3: Differentiate the Outer Function
First, we differentiate the outer function \( f(u) \):
\[ f'(u) = \frac{1}{2} u^{-\frac{1}{2}} = \frac{1}{2\sqrt{u}} \]

### Step 4: Differentiate the Inner Function
Next, we differentiate the inner function \( g(x) = ax + b \):
\[ g'(x) = a \]

### Step 5: Combine Using the Chain Rule
Now, we apply the chain rule:
\[ \frac{dy}{dx} = f'(g(x)) \cdot g'(x) = \frac{1}{2\sqrt{ax + b}} \cdot a \]

### Step 6: Simplify the Expression
Thus, we can simplify the expression:
\[ \frac{dy}{dx} = \frac{a}{2\sqrt{ax + b}} \]

### Final Answer
The derivative of \( \sqrt{ax + b} \) with respect to \( x \) is:
\[ \frac{dy}{dx} = \frac{a}{2\sqrt{ax + b}} \]

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