Differentiation with fractions, radicands, and the power chain rule

[SHLOMOLOGIC]

Differentiate the following two problems.

1. x divided by the square root of x squared+ 1

2. The square root of x + 2
divided by the square root of x - 1

Thank you.

 

[MarkFL]

Hello and welcome to MHB, SHLOMOLOGIC! :D

We ask that our users show their progress (work thus far or thoughts on how to begin) when posting questions. This way our helpers can see where you are stuck or may be going astray and will be able to post the best help possible without potentially making a suggestion which you have already tried, which would waste your time and that of the helper.

Can you post what you have done so far?

 

[SHLOMOLOGIC]

Hello and welcome to MHB, SHLOMOLOGIC! :D

We ask that our users show their progress (work thus far or thoughts on how to begin) when posting questions. This way our helpers can see where you are stuck or may be going astray and will be able to post the best help possible without potentially making a suggestion which you have already tried, which would waste your time and that of the helper.

Can you post what you have done so far?

Find derivative of X/sqrt (x^2 + 1))

using the quotient rule and the power chain rule

x/(x^2 + 1) ^1/2

derivative of x=1
derivative of denominator=1/2(x^2 +1)^-1/2 times 2x

quotient rule:

[( x^2 + 1)^1/2 times 1] ( minus) x times [above derivative of denominator]

all of the above line divided by the the denominator squared

I get (x^2+1)^1/2 (minus) x times ½ (x^2 + 1) ^-1/2 (times) 2x/ x^2 + 1

I then reduce this to (x^2 + 1)^-1/2 –- x^2(x^2 + 1)^--3/2

This does not match the book answer, 1/(x^2 + 1)^ 3/2

 

[MarkFL]

Okay, we are given:

$$f(x)=\frac{x}{\sqrt{x^2+1}}=x\left(x^2+1\right)^{-\frac{1}{2}}$$

If we apply the product rule, we find:

$$f'(x)=x\left(-\frac{1}{2}\left(x^2+1\right)^{-\frac{3}{2}}2x\right)+(1)\left(x^2+1\right)^{-\frac{1}{2}}=\left(x^2+1\right)^{-\frac{1}{2}}-x^2\left(x^2+1\right)^{-\frac{3}{2}}$$

This is what you got using the quotient rule. :) Now if we factor, we obtain:

$$f'(x)=\left(x^2+1\right)^{-\frac{3}{2}}\left(x^2+1-x^2\right)=\left(x^2+1\right)^{-\frac{3}{2}}$$

 

[MarkFL]

For the second problem, I think I would write:

$$f(x)=\left(\frac{x+2}{x-1}\right)^{\frac{1}{2}}$$

Then, I would apply the power/chain/quotient rules. Can you proceed?

 

[SHLOMOLOGIC]

For the second problem, I think I would write:

$$f(x)=\left(\frac{x+2}{x-1}\right)^{\frac{1}{2}}$$

Then, I would apply the power/chain/quotient rules. Can you proceed?

Thanks for responding.

First, where did you get the square root symbols and the textbook notation?

Secondly, I did try your method. I believe the idea is to, step one, use the power rule on the entire expression. Step two involves using the quotient rule to determine the derivative of the fraction. Then multiply and factor. factor.

Perhaps I made an error in my factoring and/or my square-root arithmetic; I'll certainly try it again, but I'm worn out after several hours of failed attempts yesterday.

I don't believe my result can be transformed into the book answer. I ended up with (x+2) to the one-half/(x-1) to the five-halfs--or is it halves?

 

[SHLOMOLOGIC]

Got it. Finally got my arithmetic right. It's easy when you know how.

 

[MarkFL]

To make the application of the chain rule a bit simpler, we can utilize:

$$\frac{x+2}{x-1}=1+3(x-1)^{-1}$$

And so:

$$f'(x)=\frac{1}{2}\left(\frac{x+2}{x-1}\right)^{-\frac{1}{2}}\left(-3(x-1)^{-2}\right)=-\frac{3}{2}\left(\frac{x-1}{x+2}\right)^{\frac{1}{2}}(x-1)^{-2}=-\frac{3}{2\sqrt{(x+2)(x-1)^3}}$$