Differentiate using `1^(st)` principle : `f(x)=(1)/(sqrt(2x+3))`

Answer

To differentiate the function \( f(x) = \frac{1}{\sqrt{2x + 3}} \) using the first principle of derivatives, we will follow these steps:

### Step 1: Write the definition of the derivative
The derivative of a function \( f \) at a point \( a \) using the first principle is given by:
\[ f'(a) = \lim_{h \to 0} \frac{f(a + h) - f(a)}{h} \]

### Step 2: Substitute the function into the definition
We need to calculate \( f(a + h) \) and \( f(a) \):
\[ f(a) = \frac{1}{\sqrt{2a + 3}} \]
\[ f(a + h) = \frac{1}{\sqrt{2(a + h) + 3}} = \frac{1}{\sqrt{2a + 2h + 3}} \]
Now, substituting these into the derivative formula:
\[ f'(a) = \lim_{h \to 0} \frac{\frac{1}{\sqrt{2a + 2h + 3}} - \frac{1}{\sqrt{2a + 3}}}{h} \]

### Step 3: Simplify the expression
To simplify the expression, we will find a common denominator:
\[ f'(a) = \lim_{h \to 0} \frac{\sqrt{2a + 3} - \sqrt{2a + 2h + 3}}{h \cdot \sqrt{2a + 2h + 3} \cdot \sqrt{2a + 3}} \]

### Step 4: Rationalize the numerator
To eliminate the square roots in the numerator, we multiply and divide by the conjugate:
\[ f'(a) = \lim_{h \to 0} \frac{(\sqrt{2a + 3} - \sqrt{2a + 2h + 3})(\sqrt{2a + 3} + \sqrt{2a + 2h + 3})}{h \cdot \sqrt{2a + 2h + 3} \cdot \sqrt{2a + 3} \cdot (\sqrt{2a + 3} + \sqrt{2a + 2h + 3})} \]
This simplifies the numerator to:
\[ (2a + 3) - (2a + 2h + 3) = -2h \]
Thus, we have:
\[ f'(a) = \lim_{h \to 0} \frac{-2h}{h \cdot \sqrt{2a + 2h + 3} \cdot \sqrt{2a + 3} \cdot (\sqrt{2a + 3} + \sqrt{2a + 2h + 3})} \]

### Step 5: Cancel \( h \)
We can cancel \( h \) from the numerator and denominator:
\[ f'(a) = \lim_{h \to 0} \frac{-2}{\sqrt{2a + 2h + 3} \cdot \sqrt{2a + 3} \cdot (\sqrt{2a + 3} + \sqrt{2a + 2h + 3})} \]

### Step 6: Apply the limit
As \( h \) approaches 0, \( \sqrt{2a + 2h + 3} \) approaches \( \sqrt{2a + 3} \):
\[ f'(a) = \frac{-2}{\sqrt{2a + 3} \cdot \sqrt{2a + 3} \cdot (2\sqrt{2a + 3})} = \frac{-2}{(2a + 3) \cdot (2\sqrt{2a + 3})} \]

### Step 7: Final expression
Thus, we can simplify this to:
\[ f'(a) = -\frac{1}{(2a + 3)^{3/2}} \]

### Conclusion
Therefore, the derivative of the function \( f(x) = \frac{1}{\sqrt{2x + 3}} \) is:
\[ f'(x) = -\frac{1}{(2x + 3)^{3/2}} \]