How do you find the derivative of a function involving square roots?

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Discussion Overview

The discussion revolves around finding the derivatives of functions involving square roots, specifically focusing on functions like f(x) = sqrt(1 - x²) and sqrt(x² + y²). Participants explore differentiation techniques, including the application of the chain rule and partial derivatives.

Discussion Character

  • Technical explanation
  • Mathematical reasoning
  • Homework-related

Main Points Raised

  • One participant asks how to differentiate f(x) = sqrt(1 - x²).
  • Another participant suggests writing the square root in exponent form as (1 - x²)^(1/2).
  • A response indicates that applying differentiation rules is necessary, but a participant's initial attempt at differentiation is critiqued for errors in applying the chain rule.
  • Another participant presents a similar problem involving sqrt(x² + y²) and expresses confusion about the differentiation process.
  • Clarification is provided regarding partial derivatives, suggesting treating one variable as constant while differentiating with respect to the other.
  • One participant confirms their differentiation steps for both variables, receiving positive feedback on their approach.
  • A new problem involving 4/^(5sqrt(x^5)) is introduced, with a suggestion to rewrite it for easier differentiation.

Areas of Agreement / Disagreement

Participants generally agree on the methods of differentiation discussed, but there are varying levels of understanding and application of the chain rule and partial derivatives. Some confusion remains regarding specific steps in the differentiation process.

Contextual Notes

Some participants express uncertainty about the correct application of differentiation rules, particularly in the context of the chain rule and partial derivatives. There are also unresolved mathematical steps in the differentiation processes presented.

Who May Find This Useful

Students or individuals seeking assistance with calculus, particularly in differentiating functions involving square roots and applying the chain rule and partial derivatives.

the1024b
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How can i find the derivative of a function like this:
f(x) = sqrt( 1 - x² )
 
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Do you know how to write a square root with exponents?
 
(1 - x² )^(1/2) ?
 
That's right! Now, you just need to apply what you know about differentiating expressions like that.
 
si will that be:
1/2((1-x²)/2)^(-1/2)

?
 
Not quite. You have one too many "1/2"s (you don't want that "/2" inside the square root and you didn't use the chain rule.

You need to multiply by the derivative of 1-x2.
 
Last edited by a moderator:
Hi,

I have a similar problem, I need to differentiate sqrt(x^2 + y^2) in terms of x and y.
Starting this I took the simple step (x^2 + y^2)^(1/2)...

My next step is a guess and I am lost after it...(1/2)(x^2 + y^2)(-1/2)...

Any help would be much appreciated.
 
mathsn00b said:
Hi,

I have a similar problem, I need to differentiate sqrt(x^2 + y^2) in terms of x and y.
Starting this I took the simple step (x^2 + y^2)^(1/2)...

My next step is a guess and I am lost after it...(1/2)(x^2 + y^2)(-1/2)...

Any help would be much appreciated.

If by "in terms of x and y", you mean you want to calculate the partial derivatives, then for the partial derivative with respect to x, treat y as a constant and differentiate with respect to x as you normally would a function of one variable. For the partial derivative with respect to y, treat x as constant.
 
thanks, would I do this by...

df/dx = 1/2(x^2 + y^2)^(-1/2).2x = x/sqrt(x^2 + y^2) and...

df/dy = 1/2(x^2 + y^2)^(-1/2).2y = y/sqrt(x^2 + y^2) ?

thanks for your help so quickly.
 
  • #10
mathsn00b said:
thanks, would I do this by...

df/dx = 1/2(x^2 + y^2)^(-1/2).2x = x/sqrt(x^2 + y^2) and...

df/dy = 1/2(x^2 + y^2)^(-1/2).2y = y/sqrt(x^2 + y^2) ?

thanks for your help so quickly.

Looks good to me.
 
  • #11
What if i have a problem similar to these however now its 4/ ^5sqrt(x^5)
 
  • #12
68Pirate said:
What if i have a problem similar to these however now its 4/ ^5sqrt(x^5)
If that is meant to be 4^(5(sqrt(x^5))), then you can easily rewrite this to equal
4^(5(x^(5/2)) And using what you know from differentiating exponentials and chain rule, you should be able to get the rest.
 

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