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Question: How is the differentiation of $xy=constant$ equal to $x\text{d}y+y\text{d}x$?

My Approach: I first tried using partial differentiation, which I know very little of. Basically, it's the differentiation of the function with respect to one variable at a time, while keeping the other constant, right?

So using that, shouldn't I get the answer as $x+y$?

All help will be appreciated greatly.

General Kenobi
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  • This is an application of the total derivative. – log_math Jan 25 '21 at 17:25
  • Think about the product rule, $\frac{d}{dx}uv = u'v + v'u$. – Joshua Wang Jan 25 '21 at 17:30
  • @JoshuaWang In your example, $uv=xy$ and then the $x$ in $\frac{d}{dx}uv$ would be equal to some other variable $z$, right? Now using this,I'd get $\frac{d}{dz}x+ \frac{d}{dz}y$. What will happen to the $dz$? – General Kenobi Jan 25 '21 at 17:36
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    Think about the word "differentiation" we are using it here in a loose sense; as in not referring to derivatives with respect to a certain variable; rather variations in the LHS and variations in the RHS (which is zero). With that in mind Joshua Wang's answer should make more sense. – Bertrand Einstein IV Jan 25 '21 at 17:39

2 Answers2

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In general, the differential of the function $f(x,y)$ is given by $$ df=f_x(x,y)dx+f_y(x,y)dy $$ where $f_x$ and $f_y$ denote the partial derivatives.

In your example, if you set $f(x,y)=xy$, you have $$ df=ydx+xdy $$

Notes.

The expression $xy=C$ is not a function, but an equation. You do not differentiate an equation per se, although you can differentiate both sides of an equation.

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Consider a function in a single variable only, say for example $f(x)=xsinx$. How would you go about finding $f'(x)?$ Certainly, you would have the privilege of using chain rule to differentiate $f(x)$. Try to think of using the same chain rule in case of $f(x,y)=xy$.

Note: xy can't be a function in a single variable, if you assume both x and y as variables.

Assume $y$ to be some function of $x$. Just as you differentiated $xsinx$, try to differentiate $xy$ now(with respect to $x$). You can work it out and see that- $$x\frac{dy}{dx}+y=0$$ and then, you can reach your coveted equation. If you wanna look up what $dy$ and $dx$ really are, then go for differentials and read it up. It's fairly a good read. As of partial differential, get to know what a differential actually is, and apply your known concepts in 1 variable differentiation to understand it.

PS: The equation which you got is a "differential equation". This can be solved via integration in order to come to your f(x,y)=xy

Coming to the last question, apply the previous concepts again and differentiate with respect to x to get it done! More can be learnt after you go on to deal with multi-variable calculus.

Abhinav Tahlani
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