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September 1st, 2016, 05:23 AM  #1 
Newbie Joined: Sep 2016 From: York Posts: 2 Thanks: 0  Quick differentiation problem
Hi, I'm Josh, just had a problem I've been thinking about and I think I'm missing a piece of... so... It's a bit of implicit differentiation, I just need to know how to implicitly differentiate half of an equation. Which is sqrt(x+y), differentiating with respect to x. obviously ( sqrt of x ) prime is 1/2(x)^.5, I just was confused about the chain rule does it come into play at all here? With the y? And so... thanks. 
September 1st, 2016, 07:12 AM  #2 
Senior Member Joined: Sep 2015 From: USA Posts: 2,529 Thanks: 1389 
$\dfrac {d}{dx} (x+y) = 1$ $\dfrac {d}{dy} (x+y) = 1$ so in both cases the factor caused by the application of the chain rule is simply 1. 
September 1st, 2016, 11:26 AM  #3 
Newbie Joined: Sep 2016 From: York Posts: 2 Thanks: 0 
I want to know why though. Like what happens to the dy? Is it just not there at all?

September 1st, 2016, 11:33 AM  #4 
Senior Member Joined: Dec 2013 From: some subspace Posts: 212 Thanks: 72 Math Focus: real analysis, vector analysis, numerical analysis, discrete mathematics 
If we have $\displaystyle y = y(x)$ and we have some function, say $\displaystyle x^2 + y^2 = 9$ (a circle), then the implicit derivative of this with respect to $\displaystyle x$ is: $\displaystyle 2x + 2yy' = 0$. $\displaystyle 2x$ is the derivative of $\displaystyle x^2$ $\displaystyle 2yy'$ is the derivative of $\displaystyle y^2 = y(x)^2$ by using the chain rule. $\displaystyle 0$ is the derivative of $\displaystyle 9$. A warning, though, implicit differentiation can be dangerous if you don't know whether or not the $\displaystyle y$ really is a function. 

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