Find the derivatives of the following :
`G(x)=(sqrt(2)x^(3)+x^(5))(sqrt(3)x^(2)+1/5x^(5))`

Answer

To find the derivative of the function \( G(x) = (\sqrt{2}x^3 + x^5)(\sqrt{3}x^2 + \frac{1}{5}x^5) \), 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:

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

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

### Step 2: Find the derivatives \( u' \) and \( v' \)
1. **Differentiate \( u(x) \)**:
\[ u'(x) = \frac{d}{dx}(\sqrt{2}x^3) + \frac{d}{dx}(x^5) = 3\sqrt{2}x^2 + 5x^4 \]

2. **Differentiate \( v(x) \)**:
\[ v'(x) = \frac{d}{dx}(\sqrt{3}x^2) + \frac{d}{dx}\left(\frac{1}{5}x^5\right) = 2\sqrt{3}x + x^4 \]

### Step 3: Apply the product rule
Now we apply the product rule:
\[ G'(x) = u'v + uv' \]

Substituting \( u, u', v, \) and \( v' \):
\[ G'(x) = (3\sqrt{2}x^2 + 5x^4)(\sqrt{3}x^2 + \frac{1}{5}x^5) + (\sqrt{2}x^3 + x^5)(2\sqrt{3}x + x^4) \]

### Step 4: Expand the expression
1. **First term**:
\[ (3\sqrt{2}x^2)(\sqrt{3}x^2) + (3\sqrt{2}x^2)(\frac{1}{5}x^5) + (5x^4)(\sqrt{3}x^2) + (5x^4)(\frac{1}{5}x^5) \]
\[ = 3\sqrt{6}x^4 + \frac{3\sqrt{2}}{5}x^7 + 5\sqrt{3}x^6 + x^9 \]

2. **Second term**:
\[ (\sqrt{2}x^3)(2\sqrt{3}x) + (\sqrt{2}x^3)(x^4) + (x^5)(2\sqrt{3}x) + (x^5)(x^4) \]
\[ = 2\sqrt{6}x^4 + \sqrt{2}x^7 + 2\sqrt{3}x^6 + x^9 \]

### Step 5: Combine like terms
Now combine all the terms:
\[ G'(x) = (3\sqrt{6} + 2\sqrt{6})x^4 + (5\sqrt{3} + 2\sqrt{3})x^6 + \left(\frac{3\sqrt{2}}{5} + 1\right)x^7 + 2x^9 \]
\[ = 5\sqrt{6}x^4 + 7\sqrt{3}x^6 + \left(\frac{3\sqrt{2}}{5} + 1\right)x^7 + 2x^9 \]

### Final Answer
Thus, the derivative \( G'(x) \) is:
\[ G'(x) = 5\sqrt{6}x^4 + 7\sqrt{3}x^6 + \left(\frac{3\sqrt{2}}{5} + 1\right)x^7 + 2x^9 \]