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?


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Kav10 Kav10
  • nice thanks

  • @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 Kav10

      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 Kav10

      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.

  • @Kav10 thx, now it's clear.

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