# proof of $d(xy) = x(dy) + y(dx)$

I was trying to prove

$$d(xy) = x(dy) + y(dx)$$

earlier this morning and I used this post to help me understand the task.

I understood the entirety of the post for my calculus class, apart from one step.

Considering an area of a rectangle with dimensions $$x$$ and $$y$$ $$xy$$ makes sense.

Likewise, the area of a rectangle with dimensions $$(x+\Delta x)(y+\Delta y)$$ giving

$$A_1=(x+\Delta x)(y+\Delta y)=xy+x \Delta y+y\Delta x+\Delta x \Delta y$$

I was confused however about the subtraction of the two areas that gives

$$x \Delta y+y\Delta x+\Delta x \Delta y$$

but the small approximation of $$\Delta x$$ and $$\Delta y$$ very small, ensuring that $$\Delta x \Delta y$$ is negligible afterwards made sense.

Why is there the need to subtract the areas as part of the proof for this differentiation property?

• You are concerned with the change in area of the rectangle, i.e., the new area minus the old area. Hence you subtract $xy$.
– kccu
Commented Mar 31, 2019 at 17:34
• By definition, $\Delta(xy)=(x+\Delta x)(y+\Delta y)-xy$, and $d(xy)$ is the linear approximation of $\Delta(xy)$. Commented Mar 31, 2019 at 17:39
• $xy$ is the current area. If you are concerned with the change, you look only at the terms with $\Delta$. Once we talk about infinitesimal changes, we get differentials. Commented Mar 31, 2019 at 18:14
• $dx$ is not a rigorous quantity Commented Jun 26, 2021 at 9:03

By $$d(xy)$$ we mean change in xy, Remember when a differential was defined, it was "Infinitesimal Change".

• Wikipedia says:

What exactly is change? Roughly speaking Final - Initial.

The initial here is $$xy$$.

We wish to express final in terms of change in the individual variables $$x$$ and $$y$$.

Hence we do $$d(xy) = (x+dx)(y+dy) - xy \ \ \ [Final \ - \ Initial]$$

Whenever dealing with derivatives change is an inevitable part. Now , what the person was trying to explain is let $$A=x.y$$ and now he was trying to find the CHANGE In Area $$A$$ for some infinitesimal small change in an arbitrary variable z. Mathematically that means $$\frac{dA}{dz}$$ which is similar to $$\frac{\Delta A}{\Delta z}$$ when the change in $$\Delta z$$ is not small and as we know $$\Delta A$$ means change in area of triangle i.e $$A_0 -A_1$$ which is what is done in the post .

Now , $$\frac{dA}{dz}=\frac{d(x.y)}{dz}= x.\frac{dy}{dz}+y.\frac{d x}{dz}$$(product rule) Taking ,$$dz$$ common from both sides and eliminating gives $$d(x.y)=x.dy+y.dx$$