× Didn't find what you were looking for? Ask a question
  
  
  
  
Top Posters
Since Sunday
39
14
10
9
9
8
7
7
7
7
6
6
New Topic  
wrote...
Administrator
Educator
Posts: 24713
10 months ago
Set:

\(f\left(x\right)=\sqrt[3]{x^2}\)

Apply into limit expression, do not take the limit yet:

\(\lim _{x\rightarrow 0}\ \frac{\sqrt[3]{\left(x+\Delta x\right)^2}-\sqrt[3]{x^2}}{\Delta x}\)

Set a and b:

\(a=\sqrt[3]{\left(x+\Delta x\right)^2}\)

Solve for a3:

\(a^3=\left(x+\Delta x\right)^2\)

Next, we look at b:

\(b=\sqrt[3]{x^2}\)

Solve for b3:

\(b^3=x^2\)

Recall:

\(a^3-b^3=\left(a-b\right)\left(a^2+ab+b^2\right)\)

Rearrange this formula:

\(\frac{a^3−b^3}{(a^2+ab+b^2)}=a−b\)

\(\frac{\left(x+\Delta x\right)^2−x^2}{\left(\sqrt[3]{\left(x+\Delta x\right)^4}\right)+\sqrt[3]{\left(x+\Delta x\right)^2x^2}+\sqrt[3]{x^4}}=a−b\)

We expand the numerator:

\(\left(x+\Delta x\right)^2−x^2=2\Delta x\cdot x+\Delta x^2\)

Common factor it further!

\(2\Delta x\cdot x+\Delta x^2=\Delta x\left(2x+\Delta x\right)\)

Apply into limit expression, do not take the limit yet:

\(\lim _{x\rightarrow 0}\frac{\Delta x\left(2x+\Delta x\right)}{\left[\left(\sqrt[3]{\left(x+\Delta x\right)^4}\right)+\sqrt[3]{\left(x+\Delta x\right)^2x^2}+\sqrt[3]{x^4}\right]\Delta x}\)

The Δx cancel each other out:

\(\lim _{x\rightarrow 0}\frac{2x+\Delta x}{\left[\left(\sqrt[3]{\left(x+\Delta x\right)^4}\right)+\sqrt[3]{\left(x+\Delta x\right)^2x^2}+\sqrt[3]{x^4}\right]}\)

Take the limit now:

\(=\frac{2x}{\left(\sqrt[3]{x^4}\right)+\sqrt[3]{x^4}+\sqrt[3]{x^4}}\)

\(=\frac{2x}{3\left(\sqrt[3]{x^4}\right)}\)

Technically you're done, but you can use the laws of exponents to go further:

\(=\frac{2x}{3\left(\sqrt[3]{x^4}\right)}\rightarrow \frac{2}{3}\cdot \frac{x}{x^{\frac{4}{3}}}\rightarrow \ \frac{2}{3}x^{1-\frac{4}{3}}\rightarrow \ \frac{2}{3}x^{-\frac{1}{3}}=\frac{2}{3\sqrt[3]{x}}\)

Final answer:

\(\frac{2}{3\sqrt[3]{x}}\)
Source  First principles, Calculus
Read 183 times
The best way to say thank you is with a positive review:

  https://trustpilot.com/review/biology-forums.com 

Your support goes a long way!


Make a note request here
Related Topics
New Topic      
Hold tight!  A Bio Forums Expert has been contacted to start answering this thread.
Explore
Post your homework questions and get free online help from our incredible volunteers.
Learn More
Improve Grades
Help Others
Save Time
Accessible 24/7
  133 People Browsing
Your Opinion
Do you believe in global warming?
Votes: 182

Related Images
 128
 133
 21

▶️ Video: Hip Animation

For a complete list of videos, visit our video library