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September 1st, 2016, 05:23 AM   #1
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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...
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September 1st, 2016, 07:12 AM   #2
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$\dfrac {d}{dx} (x+y) = 1$

$\dfrac {d}{dy} (x+y) = 1$

so in both cases the factor caused by the application of the chain rule is simply 1.
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September 1st, 2016, 11:26 AM   #3
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I want to know why though. Like what happens to the dy? Is it just not there at all?
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September 1st, 2016, 11:33 AM   #4
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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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