Differentiate the following with respect to x using first principle method.
`(x^(2))/(x+1)`

Answer

To differentiate the function \( f(x) = \frac{x^2}{x+1} \) using the first principle method, we follow these steps:

### Step 1: Define the function
Let \( f(x) = \frac{x^2}{x+1} \).

### Step 2: Apply the first principle of differentiation
According to the first principle, the derivative \( f'(x) \) is given by:
\[ f'(x) = \lim_{h \to 0} \frac{f(x+h) - f(x)}{h} \]

### Step 3: Calculate \( f(x+h) \)
We need to find \( f(x+h) \):
\[ f(x+h) = \frac{(x+h)^2}{(x+h)+1} = \frac{x^2 + 2xh + h^2}{x + h + 1} \]

### Step 4: Substitute \( f(x+h) \) and \( f(x) \) into the limit
Now we substitute \( f(x+h) \) and \( f(x) \) into the limit:
\[ f'(x) = \lim_{h \to 0} \frac{\frac{x^2 + 2xh + h^2}{x + h + 1} - \frac{x^2}{x + 1}}{h} \]

### Step 5: Find a common denominator
To simplify the expression, we find a common denominator for the two fractions:
\[ f'(x) = \lim_{h \to 0} \frac{(x^2 + 2xh + h^2)(x + 1) - x^2(x + h + 1)}{h \cdot (x + h + 1)(x + 1)} \]

### Step 6: Simplify the numerator
Expanding the numerator:
\[ = (x^2 + 2xh + h^2)(x + 1) - x^2(x + h + 1) \]
\[ = (x^3 + 2x^2h + x^2 + 2xh + h^2) - (x^3 + x^2h + x^2) \]
\[ = 2x^2h + 2xh + h^2 - x^2h \]
\[ = (2x^2 - x^2)h + 2xh + h^2 = x^2h + 2xh + h^2 \]

### Step 7: Factor out \( h \)
Now we can factor \( h \) from the numerator:
\[ = h(x^2 + 2x + h) \]

### Step 8: Substitute back into the limit
Substituting back into the limit gives:
\[ f'(x) = \lim_{h \to 0} \frac{h(x^2 + 2x + h)}{h \cdot (x + h + 1)(x + 1)} \]

### Step 9: Cancel \( h \)
We can cancel \( h \) from the numerator and denominator:
\[ f'(x) = \lim_{h \to 0} \frac{x^2 + 2x + h}{(x + h + 1)(x + 1)} \]

### Step 10: Take the limit as \( h \to 0 \)
Now we take the limit as \( h \) approaches 0:
\[ f'(x) = \frac{x^2 + 2x + 0}{(x + 0 + 1)(x + 1)} = \frac{x^2 + 2x}{(x + 1)^2} \]

### Final Answer
Thus, the derivative of \( f(x) = \frac{x^2}{x+1} \) is:
\[ f'(x) = \frac{x^2 + 2x}{(x + 1)^2} \]

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