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I'm reading Lang's First Course in Calculus 5e. I am stuck on p. 104, where he introduces Implicit Differentiation. Here's the text:

"Find the derivative $dy/dx$ if $x^2 + y^2=7$, in terms of $x$ and $y$. We differentiate both sides of the equation using the chain rule, and the fact that $dx/dx=1$. We then obtain:

$2x\frac{dx}{dx} + 2y\frac{dy}{dx}=0$, that is $2x + 2y\frac{dy}{dx}=0$

So my question is totally basic: what does $\frac{dx}{dx}$ stand for? What's it doing in there in the first place? I thought $x^2$ could be differentiated directly to $2x$, without any intermediate steps, and I'm just not seeing what $\frac{dx}{dx}$ is doing in there. Thanks for any tips.

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    I think he is just be super crystal clear with what step is occuring: he is regarding everything on the LHS as a function of x, and differentiating with the chain rule even though for that first term, its really not needed.2011-09-22
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    You are right, at this stage it causes unnecessary confusion. But if we were moving around the circle $x^2+y^2=7$, and $x$ and $y$ were both functions of time $t$, we would have $2x\frac{dx}{dt}+2y\frac{dy}{dt}=0$. However, that's for later!2011-09-22
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    I appreciate the nudges in the right direction, but I'm still head-scratching. How about this: how would you say $2x\frac{dx}{dx}$ in English? I think this is one of the things I'm really missing out on by studying on my own rather than in a class...2011-09-22

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The $\frac{dx}{dx}$ is due to the chain rule. It means what other differentials mean: the rate of change of $x$ with respect to the rate of change of... well, $x$ in this case!
Since x changes at the same rate as itself (hopefully no proof of this is needed!),
this ratio is equal to $1$
And yes, the derivative of $x^2$, with respect to $x$, is $2x$.

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    That's done it, thanks much -2011-09-22
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    You might want to check out this question (and the answers): http://math.stackexchange.com/q/21199/110202011-09-22
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    Thank you for the pointer, that question was also very illuminating!2011-09-22
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    Anytime!$$$$ :)2011-09-22