derivative of the function \( G(x)=\frac{1}{2} x^{4}+6 x^{3}-x^{2}+1.5 x+\sqrt{2} \)

To find the derivative of the function \( G(x) = \frac{1}{2} x^{4} + 6 x^{3} - x^{2} + 1.5 x + \sqrt{2} \), we can apply the power rule, which states that the derivative of \( x^n \) is \( n \cdot x^{n-1} \). Calculating the derivative term by term, we get: 1. The derivative of \( \frac{1}{2} x^{4} \) is \( 2 x^{3} \). 2. The derivative of \( 6 x^{3} \) is \( 18 x^{2} \). 3. The derivative of \( -x^{2} \) is \( -2x \). 4. The derivative of \( 1.5 x \) is \( 1.5 \). 5. The derivative of the constant \( \sqrt{2} \) is \( 0 \). Putting it all together, the derivative \( G'(x) \) is: \[ G'(x) = 2 x^{3} + 18 x^{2} - 2 x + 1.5. \]