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

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SUMMARY

The discussion focuses on finding the derivatives of functions involving square roots, specifically f(x) = sqrt(1 - x²) and sqrt(x² + y²). Participants emphasize the importance of applying the chain rule correctly and differentiating with respect to each variable when dealing with partial derivatives. The correct derivatives are df/dx = x/sqrt(x² + y²) and df/dy = y/sqrt(x² + y²). Additionally, a method for differentiating expressions like 4^(5sqrt(x^5)) is briefly mentioned, highlighting the need to rewrite the expression for clarity.

PREREQUISITES
  • Understanding of basic calculus concepts, including derivatives and the chain rule.
  • Familiarity with square root functions and their representation as exponents.
  • Knowledge of partial derivatives and how to treat constants in multivariable functions.
  • Experience with differentiating exponential functions.
NEXT STEPS
  • Study the application of the chain rule in calculus.
  • Learn how to compute partial derivatives for multivariable functions.
  • Explore differentiation techniques for exponential functions.
  • Practice finding derivatives of more complex functions involving roots and exponents.
USEFUL FOR

Students and professionals in mathematics, particularly those studying calculus, as well as educators looking to enhance their teaching methods for derivatives involving square roots and multivariable functions.

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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