Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. Join them; it only takes a minute:

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

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.

share|cite|improve this question
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. – Ragib Zaman Sep 22 '11 at 5:15
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! – André Nicolas Sep 22 '11 at 5:20
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... – Peter Sep 22 '11 at 5:29
up vote 3 down vote accepted

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 $2x$.

share|cite|improve this answer
That's done it, thanks much - – Peter Sep 22 '11 at 5:33
You might want to check out this question (and the answers): – Altar Ego Sep 22 '11 at 5:38
Thank you for the pointer, that question was also very illuminating! – Peter Sep 22 '11 at 5:49
Anytime!$$$$ :) – Altar Ego Sep 22 '11 at 13:14

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.