Differentiate the following w.r.t. x :
`x^(-1//3)e^(x)`

Answer

To differentiate the function \( y = x^{-\frac{1}{3}} e^x \) with respect to \( x \), we will use the product rule of differentiation. The product rule states that if you have two functions \( u(x) \) and \( v(x) \), then the derivative of their product is given by:

\[ \frac{d}{dx}(u \cdot v) = u \frac{dv}{dx} + v \frac{du}{dx} \]

### Step-by-Step Solution:

1. **Identify the functions**:
Let \( u = x^{-\frac{1}{3}} \) and \( v = e^x \).

2. **Differentiate \( u \)**:
To differentiate \( u \), we use the power rule:
\[ \frac{du}{dx} = -\frac{1}{3} x^{-\frac{4}{3}} \]

3. **Differentiate \( v \)**:
The derivative of \( v \) is:
\[ \frac{dv}{dx} = e^x \]

4. **Apply the product rule**:
Now, we apply the product rule:
\[ \frac{dy}{dx} = u \frac{dv}{dx} + v \frac{du}{dx} \]
Substituting the values we found:
\[ \frac{dy}{dx} = x^{-\frac{1}{3}} e^x + e^x \left(-\frac{1}{3} x^{-\frac{4}{3}}\right) \]

5. **Simplify the expression**:
We can factor out \( e^x \):
\[ \frac{dy}{dx} = e^x \left( x^{-\frac{1}{3}} - \frac{1}{3} x^{-\frac{4}{3}} \right) \]

6. **Combine the terms**:
To combine the terms inside the parentheses, we can write:
\[ \frac{dy}{dx} = e^x \left( \frac{3x^{-\frac{1}{3}} - x^{-\frac{4}{3}}}{3} \right) \]
This can be further simplified to:
\[ \frac{dy}{dx} = \frac{e^x}{3} \left( 3x^{-\frac{1}{3}} - x^{-\frac{4}{3}} \right) \]

### Final Answer:
\[ \frac{dy}{dx} = \frac{e^x}{3} \left( 3x^{-\frac{1}{3}} - x^{-\frac{4}{3}} \right) \]