# Suppose $y$, Does $\frac{dy}{dy}$ have meaning when we derive it with respect to itself?

Suppose we have a function and want to derive it with respect to itself e.g: $$\frac{dy}{dy}$$

Does this have any meaning , and if so what will be it's value?

• The answer is just 1! – danimal May 8 '15 at 10:02
• No, definitely 1 – danimal May 8 '15 at 10:04
• @GitGud No, it's 1 – Akiva Weinberger May 8 '15 at 10:48
• @AlexandreHalm The same theory in which $dy/dx$ or $dx/dx$ do. – Akiva Weinberger May 8 '15 at 11:53
• Just curious, but any particular reason for the rollback of my edit? (However, if you do not wish to explain, please feel free to ignore this comment.) – J W May 9 '15 at 6:22

We are basically asking what is the rate by which $y$ changes with respect to $y$? Since $y$ changes proportionately to itself, the value is $1$.

Notice that you probably do this implicitly if you have differentiated some functions before. For instance, if $y=2x$, then $\displaystyle\frac{d}{dx}(y)=\frac{d}{dx}(2x)=2.$ That is, the rate by which $y$ changes with respect to $x$ is $2$, since every time $y$ increases or decreases by $2$, $x$ only increases or decreases by $1$. Now suppose instead that $y=x$, since $y$ is changing only as much as $x$ is, the rate by which $y$ changes with respect to $x$ is now $1$. Thus, $\displaystyle\frac{d}{dx}(y)=\frac{d}{dx}(x)=1$.

• Possibly wrong if dy is zero (y is a constant). I am not sure what the most useful and appropriate definition to give dy/dy when y is constant is though. – Steven Stewart-Gallus May 8 '15 at 17:26
• If $y$ is meant to be a variable, how can it be constant? It would be of no use in differentiation. – yellowquark May 8 '15 at 18:54
• Basically writing $\frac{dx}{dx}$ is an abuse of notation. When writing $\frac{dx}{dx}$ what you actually mean is $\frac{df}{dx}$ when $f=\text{identity}$, i.e., $f(x)=x$. – hjhjhj57 May 8 '15 at 21:56
• @hjhjhj57 If you want to take this stance, then even writing $\frac{dy}{dx}$ is an abuse of notation, because it is nowhere indicated that $y$ is a function of $x$. The most "correct" way would be if you indicate that $D$ is an operator on functions, i.e. something like $D[(x\in\Bbb R\mapsto x)]=(x\in\Bbb R\mapsto 1)$, but needless to say this notation isn't likely to catch on anytime soon. – Mario Carneiro May 9 '15 at 8:10

Define the function $f(y)=y$ $\;\;\;\;\;\;$ (1)

We will use the first principle here,

Let a small increment in $y$ correspond to $f(y+\Delta y)$

Then , $f(y+\Delta y)=y+\Delta y$ $\;\;\;\;\;\;\;$ (2)

Subtract (1) from (2)

$f(y+\Delta y)-f(y)=\Delta y$

Re-arranging.

$\frac{f(y+\Delta y)-f(y)}{\Delta y}=1$

Taking limit $\Delta y\to 0$

$\frac{d[f(y)]}{dy}=1=\frac{d(y)}{dy}$

Hence, essentially you are measuring the rate of change of quantity with respect to itself. It's quite easy to see why it makes sense.

• Plus 1 for first principles – danimal May 8 '15 at 15:28
• What if y is a constant and so Δy equals zero? – Steven Stewart-Gallus May 8 '15 at 17:28
• If $y$ is constant, then you have a difficulty defining $d(\mbox{anything})/dy$. Every derivative has this problem; it is nothing special about $dy/dy$. – David K May 8 '15 at 21:35

$$\frac d{dy}$$ means differentiation with respect to $y$. Thus, $$\frac{dy}{dy}$$ is differentiating $y$ with respect to $y$. The derivative of $y$ with respect to $y$ is $\boxed1$. (It's slightly easier to understand if you replace all of the $y$s with $x$ — we're more used to differentiating with respect to $x$ than with respect to $y$.)

• What if y is a constant and so Δy equals zero? – Steven Stewart-Gallus May 8 '15 at 17:29
• @StevenStewart-Gallus You could say the same thing to $\frac{dy}{dx}$. What if $x$ is constant? – Akiva Weinberger May 8 '15 at 17:35
• I agree with colombus: Why would one ever differentiate anything, constant or otherwise, with respect to a constant? Objection overruled. – pjs36 May 8 '15 at 17:58

Yes , example y=y, dy/dy=1 , means that y is equal to its own coordinate and its rate of change is 1

• what does this mean in the context of the problem?? – Bak1139 May 8 '15 at 10:05
• Ignore it misread question – talmudist May 8 '15 at 10:07
• @talmudist You can delete answers, אחי. – Akiva Weinberger May 8 '15 at 11:18
• How can I delete answer ? No option for it – talmudist May 8 '15 at 11:57
• @talmudist underneath the text of your post, you should see text links that say 'share' 'edit' 'delete' 'flag' – AakashM May 8 '15 at 13:18

dy/dy is 1 unless possibly y is a constant or has constant portions. However, this statement seems kind of meaningless. What is y a constant in respect to?

One possible way to solve the problem might be to formulate everything as varying according to a single global time parameter t. Then dy/dy is better described as dyₜ/dyₜ and by the chain rule is dyₜ/dt * dt/dy. dyₜ/dt is simply 0 when y is constant with respect to 0 but finding a useful definition for dt/dyₜ is harder. As yₜ does not vary dt/dyₜ is undefined or approaches a positive infinity. As such, dyₜ/dyₜ is best described as being undefined, I think. Alternatively it is not appropriate to use the chain rule when the denominator is a constant value. However, some people may find it useful to patch in an ugly hack to our normal definition of the derivative and arbitrarily define dyₜ/dyₜ to be 1 or some other useful value for their purposes. I am not necessarily convinced such a patch would be useful but I would be open to the possibility.

• Possibly wrong if the global time parameter t that you used is constant. Also what does "when y is constant with respect to 0" mean? I think the piece you are missing is that Leibnitz notation implicitly uses the quantity that appears in the denominator as a variable. So the worry about what happens when y is constant is a red herring. As others have pointed out you would have the same problem with dy/dx worrying about what happens when x is constant as well. – BSteinhurst May 8 '15 at 19:18