Can we derive any equation

my question can we derive any equation for example

$\displaystyle \sqrt{x+y} = \sqrt{y} + \sqrt{x} $

this equation is the positive x-axis and the positive y-axis
I do not want to make it more complicated

in other word if we have a question that asks us to find $\displaystyle \frac{dy}{dx} $ should we first look if this is a curve or dose it have points on it ??
is there any continuous function that is not differentiable at all point of the domain ?

Thanks

Yes there is. $\displaystyle |X-a|$ is continuous for all x, but is not differentiable at $\displaystyle a$, where $\displaystyle a $ is any real number.

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Originally Posted by Amer

my question can we derive any equation for example

$\displaystyle \sqrt{x+y} = \sqrt{y} + \sqrt{x} $

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For this particular problem, use "implicit differentiation"
$\displaystyle (1/2)(x+ y)^{-1/2}+ (1/2)(x+ y)^{-1/2)y'= (1/2)y^{-1/2}y'+ (1/2)x^{-1/2}$
$\displaystyle \left((x+ y)^{-1/2}- y^{1/2}\right)y'= x^{1/2}- (x+y)^{-1/2}$
$\displaystyle y'= \frac{x^{-1/2}- (x+y)^{-1/2}}{(x+ y)^{-1/2}- y^{-1/2}}$

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this equation is the positive x-axis and the positive y-axis
I do not want to make it more complicated

in other word if we have a question that asks us to find $\displaystyle \frac{dy}{dx} $ should we first look if this is a curve or dose it have points on it ??

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Do you mean something like $\displaystyle x^2+ y^2= -1$?
Well, if the problem asks you to find dy/dx, it would be a very poor question if there were no points!

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is there any continuous function that is not differentiable at all point of the domain ?

Thanks

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As sambit said, y= |x- a|, for any real number a, is continuous for all x but not differentiable at x= a. There even exist functions that are continuous for all x but not differentiable for any x but they are difficult to write.

Quote:

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Originally Posted by Amer

my question can we derive any equation for example

$\displaystyle \sqrt{x+y} = \sqrt{y} + \sqrt{x} $

this equation is the positive x-axis and the positive y-axis
I do not want to make it more complicated

in other word if we have a question that asks us to find $\displaystyle \frac{dy}{dx} $ should we first look if this is a curve or dose it have points on it ??
is there any continuous function that is not differentiable at all point of the domain ?

Thanks

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"is there any continuous function that is not differentiable at all point of the domain ?" If my math history is correct, I think it was Weierstrass who first derived a continuous function that's not differentiable at any of its points!

The equation you offered also works at the origin just to remind you. And it's always good to test out the equation looking for points associated with the maxima/minima of the curve and other properties to help distinguish it

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Originally Posted by Sambit

Yes there is. $\displaystyle |X-a|$ is continuous for all x, but is not differentiable at $\displaystyle a$, where $\displaystyle a $ is any real number.

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but it is differentiable at any x not equal a

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Originally Posted by HallsofIvy

For this particular problem, use "implicit differentiation"
$\displaystyle (1/2)(x+ y)^{-1/2}+ (1/2)(x+ y)^{-1/2)y'= (1/2)y^{-1/2}y'+ (1/2)x^{-1/2}$
$\displaystyle \left((x+ y)^{-1/2}- y^{1/2}\right)y'= x^{1/2}- (x+y)^{-1/2}$
$\displaystyle y'= \frac{x^{-1/2}- (x+y)^{-1/2}}{(x+ y)^{-1/2}- y^{-1/2}}$


Do you mean something like $\displaystyle x^2+ y^2= -1$?
Well, if the problem asks you to find dy/dx, it would be a very poor question if there were no points!


As sambit said, y= |x- a|, for any real number a, is continuous for all x but not differentiable at x= a. There even exist functions that are continuous for all x but not differentiable for any x but they are difficult to write.

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Thanks I know how to drive it, I get it thanks very much