How to take the Partial Derivatives of a Function that is Defined Implicitly?

[jaguar7]

How does one take the partial derivatives of a function that is defined implicitly? For example, the function, x^2 / 4 + y^2 + z^2 = 3.

 

[HallsofIvy]

To take the partial derivatives of a function, you first have to specify the function!
$$\frac{x^2}{4}+ y^2+ z^2= 3$$
is not a function, it is an equation. If you mean that equation defines z implicitely as a function of x and y, then, differentiating with respect to x, $x/2+ 2z z_x= 0$ so that $z_x= -x/4z$ and $2y+ 2z z_y= 0$ so that $z_y= -y/z$. Notice that, in the first case, the derivative of y with respect to x is 0 and, in the second, the derivative of x with resepect to y is 0. That is because they are independent variables.

Of course, we could just as easily think of that equation as defining y in terms of the variables x and z and have $x/2+ 2yy_x= 0$ and $2y y_z+ 2z= 0$ or we could think of it as defining x in terms of y and z so that $(x/2)x_y+ 2y= 0$ and $(x/2)x_z+ 2z= 0$.