Differentiate the following functions with respect to `x` from first principles: `sqrt(a x+b)`

Options

A. `a/(2sqrt(a x+b))`
B. `a/(sqrt(a x+b))`
C. `1/(2sqrt(a x+b))`
D. `a/(2sqrt(a x-b))`

Answer

To differentiate the function \( f(x) = \sqrt{ax + b} \) from first principles, we will follow the definition of the derivative.

### Step-by-Step Solution:

1. **Define the Function**:
Let \( f(x) = \sqrt{ax + b} \).

2. **Apply the Definition of Derivative**:
According to the first principles of derivatives, the derivative \( f'(x) \) is given by:
\[ f'(x) = \lim_{h \to 0} \frac{f(x + h) - f(x)}{h} \]

3. **Substitute \( f(x + h) \)**:
We need to calculate \( f(x + h) \):
\[ f(x + h) = \sqrt{a(x + h) + b} = \sqrt{ax + ah + b} \]

4. **Set Up the Limit**:
Substitute \( f(x + h) \) and \( f(x) \) into the derivative formula:
\[ f'(x) = \lim_{h \to 0} \frac{\sqrt{ax + ah + b} - \sqrt{ax + b}}{h} \]

5. **Multiply by the Conjugate**:
To simplify the expression, multiply the numerator and the denominator by the conjugate of the numerator:
\[ f'(x) = \lim_{h \to 0} \frac{\left(\sqrt{ax + ah + b} - \sqrt{ax + b}\right) \left(\sqrt{ax + ah + b} + \sqrt{ax + b}\right)}{h \left(\sqrt{ax + ah + b} + \sqrt{ax + b}\right)} \]

6. **Simplify the Numerator**:
The numerator simplifies to:
\[ (\sqrt{ax + ah + b})^2 - (\sqrt{ax + b})^2 = (ax + ah + b) - (ax + b) = ah \]
Therefore, we have:
\[ f'(x) = \lim_{h \to 0} \frac{ah}{h \left(\sqrt{ax + ah + b} + \sqrt{ax + b}\right)} \]

7. **Cancel \( h \)**:
The \( h \) in the numerator and denominator cancels out:
\[ f'(x) = \lim_{h \to 0} \frac{a}{\sqrt{ax + ah + b} + \sqrt{ax + b}} \]

8. **Evaluate the Limit**:
As \( h \) approaches 0, \( \sqrt{ax + ah + b} \) approaches \( \sqrt{ax + b} \):
\[ f'(x) = \frac{a}{\sqrt{ax + b} + \sqrt{ax + b}} = \frac{a}{2\sqrt{ax + b}} \]

9. **Final Result**:
Therefore, the derivative of the function is:
\[ f'(x) = \frac{a}{2\sqrt{ax + b}} \]