Partial Differentiation

The del operator contains partial differential operators. Partial differentiation is when we differentiate a function of more than one variable, with respect to only one of these variables.

\( \frac{\partial f(x,y,z) }{\partial x} \) means differentiate \( f(x,y,z) \) w.r.t. \( x \) while keeping \( y \) and \( z \) constant.

Partial differentiation can be applied to surfaces. Consider the surface \( f(x,y) = x^2 + xy + y^2\) as shown below. The plane \( y = y_0 \) intersects this surface producing a curve.

The equation of this curve is given by \( f(x,y_0) = x^2 + xy_0 + y_0^2 \). This is an equation of one variable \( x \) as \( y_0 \) is constant.

What is the derivative of this new curve w.r.t. \(x\)? $$ \frac{d f(x,y_0)}{d x} = \frac{\partial f(x,y)}{\partial x} \bigg|_{y=y_0} = 2x + y_0 $$