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Apashanka

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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.

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Apashanka

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Mark44

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

Corrected, this would be $$\lim_{h \to 0}

\frac{u(x+h, y)v(x+h, y) - u(x, y)v(x,y )}{h}$$

- #3

Apashanka

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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 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}$$

- #4

Mark44

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

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

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Apashanka

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So which is correct post 2 or post 4Mark44 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 $$\lim_{h \to 0}\left(\frac{u(x+h,y)-u(x,y)}{h}\right) v(x,y)+u(x,y)\lim_{h \to 0}\left(\frac{v(x+h,y)-v(x,y)}{h}\right)$$

- #6

Mark44

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Both are correct. It's possible to derive what I wrote in post 4 from what is in post 2.Apashanka said:So which is correct post 2 or post 4

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HallsofIvy

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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.

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.

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.

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.

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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