The area of calculus that I'm questioning the difference between d/dx and dz/dx is when evaluating tangent planes and using Fx, Fy, and Fz for linear approximations.
For example, if the question asks us to evaluate the equation of a tangent plane given an equation (in R3) and a point, the first step would be to find the Fx, Fy, and Fz so we could plug it into the plane equation Fx(x-x0) + Fy(y-y0) + Fz(z-z0) = 0. Here is where my question comes in. When evaluating Fx, for example, would we use dz/dx or d/dx? What exactly is the difference between the two?
For the previous problems that I've done, they are all questions asking us to find the equation of a tangent line on a plane, and I simply used d/dx instead of dz/dx. Is this different for when evaluating the tangent plane for a 3d figure?
$\endgroup$1 Answer
$\begingroup$There are two cases :
The plane is defined by an implicit expression of the form$$F(x,y,z)=0$$
In this case, the tangent plane at the point $ (x_0,y_0,z_0) $ will have the equation you wrote :$$F_x(x-x_0)+F_y(y-y_0)+F_z(z-z_0)=0$$with$$F_x=\frac{\partial F}{\partial x}(x_0,y_0,z_0)$$and so on for $ F_y $ and $ F_z$.
The second case is when the plane is defined by an explicit expression of the form$$z=f(x,y)$$or$$G(x,y,z)=z-f(x,y)=0$$in this case, the equation of the tangent plane will be$$G_x(x-x_0)+G_y(y-y_0)+G_z(z-z_0)=$$
or
$$z-z_0=\frac{\partial f}{\partial x}(x_0,y_0)(x-x_0)+\frac{\partial f}{\partial y}(x_0,y_0)(y-y_0)$$
$$=\frac{\partial z}{\partial x}(x_0,y_0)(x-x_0)+\frac{\partial z}{\partial y}(x_0,y_0)(y-y_0)$$
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