Differentiate the following functions:
10. `(1)/( x) + (3)/( x^2) + (2)/( x^3)`

Answer

To differentiate the function \( f(x) = \frac{1}{x} + \frac{3}{x^2} + \frac{2}{x^3} \), we will follow these steps:

### Step 1: Rewrite the Function
We can rewrite the function in terms of negative exponents:
\[ f(x) = x^{-1} + 3x^{-2} + 2x^{-3} \]

### Step 2: Differentiate Each Term
Now we will differentiate each term using the power rule, which states that if \( f(x) = x^n \), then \( f'(x) = n \cdot x^{n-1} \).

1. Differentiate \( x^{-1} \):
\[ \frac{d}{dx}(x^{-1}) = -1 \cdot x^{-2} = -\frac{1}{x^2} \]

2. Differentiate \( 3x^{-2} \):
\[ \frac{d}{dx}(3x^{-2}) = 3 \cdot (-2) \cdot x^{-3} = -6x^{-3} = -\frac{6}{x^3} \]

3. Differentiate \( 2x^{-3} \):
\[ \frac{d}{dx}(2x^{-3}) = 2 \cdot (-3) \cdot x^{-4} = -6x^{-4} = -\frac{6}{x^4} \]

### Step 3: Combine the Results
Now we can combine the derivatives of each term:
\[ f'(x) = -\frac{1}{x^2} - \frac{6}{x^3} - \frac{6}{x^4} \]

### Step 4: Final Result
Thus, the derivative of the function is:
\[ f'(x) = -\frac{1}{x^2} - \frac{6}{x^3} - \frac{6}{x^4} \]