# Partial Derivative: Correct Formulation?

• I
• Apashanka
In summary, the correct way to write the partial derivative of the product of two functions is using the limit definition as shown in post 4. Using the definition of the two partial derivatives without including the limit will lead to incorrect results. Additionally, it is important to clarify the meaning of "x+ dx" as it could involve non-standard analysis.
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}##??

Apashanka said:
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 and jim mcnamara
Mark44 said:
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}??##

Apashanka said:
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
Mark44 said:
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

Apashanka said:
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.

Apashanka
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.

## 1. What is a partial derivative?

A partial derivative is a mathematical concept used in multivariable calculus to measure how a function changes with respect to one of its variables while holding the other variables constant.

## 2. How is a partial derivative different from a regular derivative?

A regular derivative measures how a function changes with respect to its independent variable, while a partial derivative measures how a function changes with respect to one of its dependent variables while holding the others constant.

## 3. What is the correct way to write a partial derivative?

The correct notation for a partial derivative is ∂f/∂x, where ∂ (the partial symbol) represents the partial derivative and f is the function being differentiated. The variable after the ∂ symbol indicates which variable is being held constant.

## 4. What is the purpose of using partial derivatives?

Partial derivatives are useful in many fields of science and engineering, such as physics, economics, and engineering, as they allow us to analyze how a system or process changes when multiple variables are involved. They are also essential in optimization problems, where we need to find the maximum or minimum value of a function with multiple variables.

## 5. Are there any rules or formulas for calculating partial derivatives?

Yes, there are several rules and formulas for calculating partial derivatives, such as the power rule, product rule, quotient rule, and chain rule. These rules are similar to those used for regular derivatives, but they take into account the other variables in the function. It is important to understand and apply these rules correctly to obtain the correct formulation of a partial derivative.

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