Find the derivatives of f(x) w.r.t. x in the following :
`f(x)=root(3)(2x^(4)+x^(2)-x)`

Answer

To find the derivative of the function \( f(x) = \sqrt[3]{2x^4 + x^2 - x} \), we will use the chain rule and the power rule of differentiation.

### Step-by-Step Solution:

1. **Rewrite the Function**:
We start by rewriting the function in a form that is easier to differentiate:
\[ f(x) = (2x^4 + x^2 - x)^{\frac{1}{3}} \]

2. **Apply the Chain Rule**:
To differentiate \( f(x) \), we apply the chain rule. The chain rule states that if you have a composite function \( g(h(x)) \), then the derivative is given by:
\[ g'(h(x)) \cdot h'(x) \]
Here, let \( g(u) = u^{\frac{1}{3}} \) and \( h(x) = 2x^4 + x^2 - x \).

3. **Differentiate the Outer Function**:
First, we differentiate \( g(u) \):
\[ g'(u) = \frac{1}{3}u^{-\frac{2}{3}} \]

4. **Differentiate the Inner Function**:
Next, we differentiate \( h(x) \):
\[ h'(x) = \frac{d}{dx}(2x^4 + x^2 - x) = 8x^3 + 2x - 1 \]

5. **Combine the Results**:
Now we combine the results using the chain rule:
\[ f'(x) = g'(h(x)) \cdot h'(x) = \frac{1}{3}(2x^4 + x^2 - x)^{-\frac{2}{3}} \cdot (8x^3 + 2x - 1) \]

6. **Final Result**:
Thus, the derivative of \( f(x) \) is:
\[ f'(x) = \frac{8x^3 + 2x - 1}{3(2x^4 + x^2 - x)^{\frac{2}{3}}} \]