To find the derivative of the function [tex]\( G(x) = \frac{1}{2} x^4 + 2 x^3 - x^2 + 5.3 x + \sqrt{10} \)[/tex], we need to apply the rules of differentiation term by term. Let's go through each step in detail:
1. Differentiate the first term:
[tex]\[
\frac{d}{dx}\left(\frac{1}{2} x^4\right)
\][/tex]
Using the power rule [tex]\(\frac{d}{dx}(x^n) = n x^{n-1}\)[/tex], we get:
[tex]\[
\frac{d}{dx}\left(\frac{1}{2} x^4\right) = \frac{1}{2} \cdot 4 x^{4-1} = 2 x^3
\][/tex]
2. Differentiate the second term:
[tex]\[
\frac{d}{dx}(2 x^3)
\][/tex]
Again, using the power rule, we get:
[tex]\[
\frac{d}{dx}(2 x^3) = 2 \cdot 3 x^{3-1} = 6 x^2
\][/tex]
3. Differentiate the third term:
[tex]\[
\frac{d}{dx}(-x^2)
\][/tex]
Using the power rule, we get:
[tex]\[
\frac{d}{dx}(-x^2) = -2 x^{2-1} = -2 x
\][/tex]
4. Differentiate the fourth term:
[tex]\[
\frac{d}{dx}(5.3 x)
\][/tex]
For a linear term [tex]\( a x \)[/tex], where [tex]\( a \)[/tex] is a constant, the derivative is just the constant [tex]\( a \)[/tex]:
[tex]\[
\frac{d}{dx}(5.3 x) = 5.3
\][/tex]
5. Differentiate the fifth term:
[tex]\[
\frac{d}{dx}(\sqrt{10})
\][/tex]
Since [tex]\(\sqrt{10}\)[/tex] is a constant, its derivative is:
[tex]\[
\frac{d}{dx}(\sqrt{10}) = 0
\][/tex]
Now, we combine all these results to find the derivative of [tex]\( G(x) \)[/tex]:
[tex]\[
G'(x) = 2 x^3 + 6 x^2 - 2 x + 5.3
\][/tex]
Thus, the derivative of the function [tex]\( G(x) \)[/tex] is:
[tex]\[
G'(x) = 2 x^3 + 6 x^2 - 2 x + 5.3
\][/tex]
