Find the derivative of `f(x)= xsin x` from the first principle.

Answer

To find the derivative of the function \( f(x) = x \sin x \) using the first principle of derivatives, we will follow these steps:

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

### Step 2: Calculate \( f(x+h) \)
Substituting \( x + h \) into the function:
\[ f(x+h) = (x+h) \sin(x+h) \]

### Step 3: Substitute \( f(x+h) \) and \( f(x) \) into the derivative formula
Now we substitute \( f(x+h) \) and \( f(x) \) into the derivative formula:
\[ f'(x) = \lim_{h \to 0} \frac{(x+h) \sin(x+h) - x \sin x}{h} \]

### Step 4: Expand \( (x+h) \sin(x+h) \)
Using the angle addition formula for sine, we can expand \( \sin(x+h) \):
\[ \sin(x+h) = \sin x \cos h + \cos x \sin h \]
Thus,
\[ f(x+h) = (x+h)(\sin x \cos h + \cos x \sin h) \]
Expanding this gives:
\[ f(x+h) = x \sin x \cos h + x \cos x \sin h + h \sin x \cos h + h \cos x \sin h \]

### Step 5: Substitute back into the limit expression
Now substituting back into our limit:
\[ f'(x) = \lim_{h \to 0} \frac{x \sin x \cos h + x \cos x \sin h + h \sin x \cos h + h \cos x \sin h - x \sin x}{h} \]

### Step 6: Simplify the expression
This simplifies to:
\[ f'(x) = \lim_{h \to 0} \frac{x \sin x (\cos h - 1) + x \cos x \sin h + h \sin x \cos h + h \cos x \sin h}{h} \]

### Step 7: Split the limit into parts
We can split the limit into separate terms:
\[ f'(x) = \lim_{h \to 0} \left( \frac{x \sin x (\cos h - 1)}{h} + \frac{x \cos x \sin h}{h} + \sin x \cos h + \cos x \sin h \right) \]

### Step 8: Evaluate the limits
1. As \( h \to 0 \), \( \cos h \to 1 \) and \( \sin h \to h \), thus:
- \( \frac{x \sin x (\cos h - 1)}{h} \to 0 \)
- \( \frac{x \cos x \sin h}{h} \to x \cos x \)
- \( \sin x \cos h \to \sin x \)
- \( \cos x \sin h \to 0 \)

Putting it all together:
\[ f'(x) = 0 + x \cos x + \sin x + 0 = x \cos x + \sin x \]

### Final Result
Thus, the derivative of \( f(x) = x \sin x \) is:
\[ f'(x) = x \cos x + \sin x \]