$Explain partial derivatives.$

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$Explain partial derivatives.$

When finding the partial derivative of a multivariable function with respect to one of its variables (let's say x), we treat the other variables (in this case, y) as constant. Does it mean that not depending on y, z(x,y) changes the same related to x for all y's?

Calculus

Babaduras

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Yes, that's right! A partial derivative of a multivariable function is its derivative with respect to one of those variables, with the others held constant. This is the definition.

When finding the partial derivative of a multivariable function with respect to one of its variables (let's say x), you treat the other variables (in this case, y) as constants. This means that you're focusing on how the function changes with respect to that specific variable while keeping the others fixed.

If you have a function f(x,y), and you're taking the partial derivative of it with respect to x while treating y as a constant, you're essentially looking at how the function f changes as you make small changes to x while keeping y fixed.

Now, regarding your question about whether f(x,y) changes the same way with respect to x for all y's, generally speaking, the partial derivative of f with respect to x can be different at different points (x, y) in the (x, y)-plane, because the function f(x,y) might respond differently to changes in x depending on the value of y.

In other words, the behavior of the partial derivative with respect to x might vary as you move around in the (x, y)-plane. So, while you treat y as constant when taking the partial derivative with respect to x, the actual value of y can influence how the function f(x,y) changes with respect to x at a particular point. In other words, the behavior of these partial derivatives can vary depending on the specific values of the variables involved.

Kav10

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Babaduras

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nice thanks
Babaduras

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@Kav10 (extra question for tip) Did I get it correctly that actual rate of change sometimes also depends on y? For example if equation for derivative involves y as here : z(x,y) = x^2y, so z_x(x,y) =2xy, so here the actual rate of change of z related to x also depends on y I choose, but for each of all possible y's, it stays the same(2xy)?
- Kav10
  
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  Yes, you've got it right. In your example, the rate of change of z with respect to x does depend on the value of y, as indicated by the term "2xy". As you correctly stated, for each specific value of y, the rate of change remains consistent. However, when you change the value of y, the rate of change will vary accordingly.
- Kav10
  
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  So, to reiterate, while the rate of change (the partial derivative) is influenced by the value of y, the pattern remains the same for all values of y. The presence of the variable y in the expression of the partial derivative indicates that the rate of change of z with respect to x is influenced by the specific combination of x and y, and this relationship remains constant as long as you are considering a fixed value of y.
Babaduras

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@Kav10 thx, now it's clear.
- Kav10
  
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  No problem!

$Explain partial derivatives.$

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