Derivative of function with radicals

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Discussion Overview

The discussion revolves around finding the derivative of the function f(x) = x + √x. Participants explore the process of applying the limit definition of the derivative and address algebraic manipulations involved in simplifying the expression.

Discussion Character

  • Homework-related
  • Mathematical reasoning

Main Points Raised

  • One participant expresses difficulty in finding the derivative and shares their progress using the limit definition, reaching a point of confusion regarding algebraic cancellation.
  • Another participant suggests that the expression can be simplified to "1 + (fraction)" and explains how to rationalize the numerator to facilitate taking the limit.
  • A participant acknowledges the help received and questions whether their earlier step was incorrect, seeking clarification on the transition to the "1 + (fraction)" form.
  • Another participant confirms that the earlier step is not wrong and provides a breakdown of how it leads to the simplified expression.

Areas of Agreement / Disagreement

Participants generally agree on the steps involved in finding the derivative, but there is some uncertainty regarding the initial algebraic manipulations and the clarity of the transition to the simplified form.

Contextual Notes

Some participants express confusion over specific algebraic steps, particularly in cancelling terms and rationalizing expressions, indicating a need for further clarification on these processes.

PhysChem
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I've been having trouble figuring out how to find the derivative of f(x) = x + √x

The farthest I got was:

[(x+h) + √(x+h) - (x+√x)]/h =

[h + √(x+h) - √x] / h

I got stuck here because I'm not sure how to cancel out h in numerator and denominator (if i can even do that at this stage) or multiply the whole equation by the conjugate. For some reason I'm stumped on what seems like simple algebra.
 
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PhysChem said:
I've been having trouble figuring out how to find the derivative of f(x) = x + √x

The farthest I got was:

[(x+h) + √(x+h) - (x+√x)]/h =

[h + √(x+h) - √x] / h

I got stuck here because I'm not sure how to cancel out h in numerator and denominator (if i can even do that at this stage) or multiply the whole equation by the conjugate. For some reason I'm stumped on what seems like simple algebra.
That is, of course, [itex]1+ \frac{\sqrt{x+h}- \sqrt{x}}{h}[/itex]. The "1" is of course, the derivative of "x". All that is left is the last fraction. To do that "rationalize" the numerator: multiply both numerator and denominator by [itex]\sqrt{x+h}+ \sqrt{x}[/itex] to get
[tex]\frac{\sqrt{x+h}- \sqrt{x}}{h}\frac{\sqrt{x+h}+\sqrt{x}}{\sqrt{x+h}+\sqrt{x}}[/tex]
[tex]= \frac{(x+ h)- x}{h(\sqrt{x+h}+\sqrt{x})}= \frac{1}{\sqrt{x+h}+ \sqrt{x}}[/tex]

Now, it's easy to take the limit as h goes to 0.
 
I see, thank you for the help! the "1 + (fraction)" was what I missed.

Does this mean my second step [h + √(x+h) - √x] / h was wrong?

I'm still not sure how to arrive at your first step, the "1 + (fraction)". However, everything else makes sense now!
 
PhysChem said:
I see, thank you for the help! the "1 + (fraction)" was what I missed.

Does this mean my second step [h + √(x+h) - √x] / h was wrong?

I'm still not sure how to arrive at your first step, the "1 + (fraction)". However, everything else makes sense now!

No your second step isn't wrong. HallsofIvy's first step comes from your second step:
[tex]\frac{h + \sqrt{x+h}- \sqrt{x}}{h}[/tex]
[tex]= \frac{h}{h} + \frac{\sqrt{x+h}- \sqrt{x}}{h}[/tex]
[tex]= 1 + \frac{\sqrt{x+h}- \sqrt{x}}{h}[/tex]
 

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