Differentiate the following functions:
`x^((5)/(3))`

Answer

To differentiate the function \( f(x) = x^{\frac{5}{3}} \), we will use the power rule of differentiation. The power rule states that if \( f(x) = x^m \), then the derivative \( f'(x) \) is given by:

\[ f'(x) = m \cdot x^{m-1} \]

### Step-by-Step Solution:

1. **Identify the exponent \( m \)**:
Here, \( m = \frac{5}{3} \).

2. **Apply the power rule**:
Using the power rule, we differentiate \( f(x) = x^{\frac{5}{3}} \):
\[ f'(x) = \frac{5}{3} \cdot x^{\frac{5}{3} - 1} \]

3. **Simplify the exponent**:
Now, simplify the exponent \( \frac{5}{3} - 1 \):
\[ \frac{5}{3} - 1 = \frac{5}{3} - \frac{3}{3} = \frac{5 - 3}{3} = \frac{2}{3} \]

4. **Substitute back into the derivative**:
Substitute the simplified exponent back into the derivative:
\[ f'(x) = \frac{5}{3} \cdot x^{\frac{2}{3}} \]

5. **Final result**:
Thus, the derivative of \( x^{\frac{5}{3}} \) is:
\[ f'(x) = \frac{5}{3} x^{\frac{2}{3}} \]

### Summary:
The derivative of the function \( x^{\frac{5}{3}} \) is \( \frac{5}{3} x^{\frac{2}{3}} \).

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