Trouble with the derivative of 1/sqrt (3x)

[awesomest47]

Hey everyone! I'm having trouble with finding the derivative of a tangent line in my calculus class. The question is for 1/sqrt (3x).

So far....
I wrote it in correct form as 1/sqrt (3x+3deltax) - 1/sqrt (3x) all over deltax
Then I combined the fractions to make sqrt(3x) - sqrt (3x+3deltax)/sqrt(3x+3deltax *sqrt (3x) over delta x
But I have no idea where to proceed...? Thanks for any help!

 

[Deleted member 4993]

Hey everyone! I'm having trouble with finding the derivative of a tangent line in my calculus class. The question is for 1/sqrt (3x).

So far....
I wrote it in correct form as 1/sqrt (3x+3deltax) - 1/sqrt (3x) all over deltax
Then I combined the fractions to make sqrt(3x) - sqrt (3x+3deltax)/sqrt(3x+3deltax *sqrt (3x) over delta x
But I have no idea where to proceed...? Thanks for any help!

\(\displaystyle \displaystyle{\dfrac{1}{\sqrt{3}}\lim_{\Delta x \to 0}\left[\dfrac{\dfrac{1}{\sqrt{(x+\Delta x)}} - \dfrac{1}{\sqrt{x}}}{\Delta x}\right ]}\)

\(\displaystyle = \ \displaystyle{\dfrac{1}{\sqrt{3}}\lim_{\Delta x \to 0}\left[\dfrac{\dfrac{\sqrt{x} -\sqrt{x+\Delta x}}{\sqrt{(x+\Delta x)} \ * \ \sqrt{x}}}{\Delta x}\right ]}\)

\(\displaystyle = \ \displaystyle{\dfrac{1}{\sqrt{3}}\lim_{\Delta x \to 0}\left[\dfrac{\sqrt{x} -\sqrt{x+\Delta x}}{\sqrt{(x+\Delta x)} \ * \ \sqrt{x} \ * \ \Delta x}\right ]}\)

\(\displaystyle = \ \displaystyle{\dfrac{1}{\sqrt{3}}\lim_{\Delta x \to 0}\left[\dfrac{\sqrt{x} -\sqrt{x+\Delta x}}{\sqrt{(x+\Delta x)} \ * \ \sqrt{x} \ * \ \Delta x} \ * \ \dfrac{\sqrt{x} + \sqrt{x+\Delta x}}{\sqrt{x} + \sqrt{x+\Delta x}}\right ]}\)

Now continue....

 

[awesomest47]

Just figured it out

Thank you for your help! I just wanted to update you and tell you that I worked through it with my study group and was able to understand the problem. Thanks for the assistance!