Partial Derivative: Correct Formulation?

[Apashanka]

If given a function $u(x,y) v(x,y)$ then is it correct to write $\frac{\partial }{\partial x}u(x,y)v(x,y)=\frac{u(x+dx,y)v(x+dx,y)-u(x,y)v(x,y)}{dx}$??

 

[Mark44]

If given a function $u(x,y) v(x,y)$ then is it correct to write $\frac{\partial }{\partial x}u(x,y)v(x,y)=\frac{u(x+dx,y)v(x+dx,y)-u(x,y)v(x,y)}{dx}$??

Not quite. Above, you're using the definition of the (partial) derivative of the product of two functions, which is a limit.

Corrected, this would be $$\lim_{h \to 0}
\frac{u(x+h, y)v(x+h, y) - u(x, y)v(x,y )}{h}$$

 

[Apashanka]

Not quite. Above, you're using the definition of the (partial) derivative of the product of two functions, which is a limit.

Corrected, this would be $$\lim_{h \to 0}
\frac{u(x+h, y)v(x+h, y) - u(x, y)v(x,y )}{h}$$

Ok using $\frac{\partial}{\partial x}u(x,y)v(x,y)=\frac{\partial u(x,y)}{\partial x}v(x,y)+u(x,y)\frac{\partial v(x,y)}{\partial x}$ can't it be $\frac{u(x+dx,y)-u(x,y)}{dx}v(x,y)+u(x,y)\frac{v(x+dx,y)-v(x,y)}{dx}??$

 

[Mark44]

Ok using $\frac{\partial}{\partial x}u(x,y)v(x,y)=\frac{\partial u(x,y)}{\partial x}v(x,y)+u(x,y)\frac{\partial v(x,y)}{\partial x}$ can't it be $\frac{u(x+dx,y)-u(x,y)}{dx}v(x,y)+u(x,y)\frac{v(x+dx,y)-v(x,y)}{dx}??$

Again, not quite -- you are trying to use the definitions of the two partial derivatives without including that these are limits.

The corrected version would be $$\lim_{h \to 0}\left(\frac{u(x+h,y)-u(x,y)}{h}\right) v(x,y)+u(x,y)\lim_{h \to 0}\left(\frac{v(x+h,y)-v(x,y)}{h}\right)$$

 

[Apashanka]

Again, not quite -- you are trying to use the definitions of the two partial derivatives without including that these are limits.

The corrected version would be $$\lim_{h \to 0}\left(\frac{u(x+h,y)-u(x,y)}{h}\right) v(x,y)+u(x,y)\lim_{h \to 0}\left(\frac{v(x+h,y)-v(x,y)}{h}\right)$$

So which is correct post 2 or post 4

 

[Mark44]

So which is correct post 2 or post 4

Both are correct. It's possible to derive what I wrote in post 4 from what is in post 2.

 

[HallsofIvy]

When you write "u(x+ dx)" exactly what do you mean by "x+ dx"? In order to make sense of that you would have to use "non-standard analysis" and I don't think that's what you mean.