Looking for reliable answers? Westonci.ca is the ultimate Q&A platform where experts share their knowledge on various topics. Discover detailed solutions to your questions from a wide network of experts on our comprehensive Q&A platform. Join our platform to connect with experts ready to provide precise answers to your questions in different areas.
Sagot :
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.
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.
We appreciate your time. Please come back anytime for the latest information and answers to your questions. We appreciate your visit. Our platform is always here to offer accurate and reliable answers. Return anytime. Get the answers you need at Westonci.ca. Stay informed by returning for our latest expert advice.