To find the derivative of the function [tex]\( g(x) = \sqrt{x^3} + 2x^2 \)[/tex], we will proceed step-by-step.
Step 1: Rewrite the function [tex]\( g(x) \)[/tex]
Rewrite [tex]\( \sqrt{x^3} \)[/tex] as [tex]\( (x^3)^{1/2} = x^{3/2} \)[/tex].
Thus,
[tex]\[ g(x) = x^{3/2} + 2x^2 \][/tex]
Step 2: Differentiate each term separately
1. Differentiating [tex]\( x^{3/2} \)[/tex]:
[tex]\[
\frac{d}{dx} x^{3/2} = \frac{3}{2} x^{(3/2) - 1} = \frac{3}{2} x^{1/2}
\][/tex]
2. Differentiating [tex]\( 2x^2 \)[/tex]:
[tex]\[
\frac{d}{dx} 2x^2 = 2 \cdot 2x = 4x
\][/tex]
Step 3: Combine the derivatives
Add the derivatives of both terms to get the derivative of the entire function:
[tex]\[
g'(x) = \frac{3}{2} x^{1/2} + 4x
\][/tex]
So, after calculating, the derivative of [tex]\( g(x) \)[/tex] is:
[tex]\[
g'(x) = \frac{3}{2} x^{1/2} + 4x
\][/tex]
Step 4: Match the correct answer choice
From the given answer choices:
a. [tex]\( g'(x) = \frac{3}{2} x^{-1/2} + 4x \)[/tex]
b. [tex]\( g'(x) = \frac{3}{2} x^{1/2} + 4x \)[/tex]
c. [tex]\( g'(x) = \frac{3}{2} x^{1/2} + 2x \)[/tex]
d. [tex]\( g'(x) = \frac{3}{2} x^{-1/2} + 2x \)[/tex]
The correct choice is (b):
[tex]\[
g'(x) = \frac{3}{2} x^{1/2} + 4x
\][/tex]
Therefore, the derivative of [tex]\( g(x) = \sqrt{x^3} + 2x^2 \)[/tex] is choice b.
