- #1

fog37

- 1,568

- 108

- TL;DR Summary
- Partial derivatives of the function f(x,y)

Hello,

Given a function like ##z= 3x^2 +2y##, the partial derivative of z w.r.t. x is equal to: $$\frac {\partial z}{\partial x} = 6x$$

Let's consider the point ##(3,2)##. If we sat on top of the point ##(3,2)## and looked straight in the positive x-direction, the slope The slope would be ##(6)(2)=12##. In taking the partial derivative, we assume that y is fixed, i.e. kept constant at ##y=2##. However, if we picked a different starting point like ##(3,4)## that has a different ##y## value, the partial derivative would still be equal to $$\frac {\partial z}{\partial x} =12 $$.

But I am envisioning a ##z## curve over the x-y plane that may have a different slope in the x-direction at the point ##(3,4)##. That seems possible. However, $$\frac {\partial z}{\partial x} = 6x$$ does not capture the fact that the local slope in the x-direction may be different at different y locations....Where am I off?

Thanks!

Given a function like ##z= 3x^2 +2y##, the partial derivative of z w.r.t. x is equal to: $$\frac {\partial z}{\partial x} = 6x$$

Let's consider the point ##(3,2)##. If we sat on top of the point ##(3,2)## and looked straight in the positive x-direction, the slope The slope would be ##(6)(2)=12##. In taking the partial derivative, we assume that y is fixed, i.e. kept constant at ##y=2##. However, if we picked a different starting point like ##(3,4)## that has a different ##y## value, the partial derivative would still be equal to $$\frac {\partial z}{\partial x} =12 $$.

But I am envisioning a ##z## curve over the x-y plane that may have a different slope in the x-direction at the point ##(3,4)##. That seems possible. However, $$\frac {\partial z}{\partial x} = 6x$$ does not capture the fact that the local slope in the x-direction may be different at different y locations....Where am I off?

Thanks!