Find the derivatives of f(x) w.r.t. x in the following :
`f(x)=(x^(2)+x+5)^(1//3)(x^(3)+1)^(2//3)`

Answer

To find the derivative of the function \( f(x) = (x^2 + x + 5)^{1/3} (x^3 + 1)^{2/3} \), 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 \( f'(x) \) is given by:

\[ f'(x) = u'v + uv' \]

### Step 1: Identify the Functions
Let:
- \( u = (x^2 + x + 5)^{1/3} \)
- \( v = (x^3 + 1)^{2/3} \)

### Step 2: Differentiate \( u \)
To differentiate \( u \), we will use the chain rule. The derivative of \( u \) is:

\[ u' = \frac{1}{3}(x^2 + x + 5)^{-2/3} \cdot (2x + 1) \]

### Step 3: Differentiate \( v \)
Now, we differentiate \( v \) using the chain rule:

\[ v' = \frac{2}{3}(x^3 + 1)^{-1/3} \cdot (3x^2) = 2x^2 (x^3 + 1)^{-1/3} \]

### Step 4: Apply the Product Rule
Now we can apply the product rule:

\[ f'(x) = u'v + uv' \]

Substituting the derivatives we found:

\[ f'(x) = \left( \frac{1}{3}(x^2 + x + 5)^{-2/3} (2x + 1) \right) (x^3 + 1)^{2/3} + (x^2 + x + 5)^{1/3} \left( 2x^2 (x^3 + 1)^{-1/3} \right) \]

### Step 5: Simplify the Expression
Now we will simplify the expression:

\[ f'(x) = \frac{1}{3}(x^2 + x + 5)^{-2/3} (2x + 1)(x^3 + 1)^{2/3} + 2x^2 (x^2 + x + 5)^{1/3} (x^3 + 1)^{-1/3} \]

### Step 6: Combine the Terms
To combine the terms, we can factor out common terms:

\[ f'(x) = \frac{1}{3} (x^2 + x + 5)^{-2/3} (x^3 + 1)^{-1/3} \left[ (2x + 1)(x^3 + 1) + 6x^2(x^2 + x + 5) \right] \]

### Final Answer
Thus, the derivative of \( f(x) \) is:

\[ f'(x) = \frac{1}{3} (x^2 + x + 5)^{-2/3} (x^3 + 1)^{-1/3} \left[ (2x + 1)(x^3 + 1) + 6x^2(x^2 + x + 5) \right] \]