Partial Differentiation

The idea

For a function of several variables like $f(x,y)$, a partial derivative with respect to one variable treats every other variable as a fixed constant, then differentiates normally:

Worked example: first partial derivatives

Find $\dfrac{\partial f}{\partial x}$ and $\dfrac{\partial f}{\partial y}$ for $f(x,y) = x^3y^2 + \sin(xy)$.

Solution: Differentiating with respect to $x$ (holding $y$ fixed, the chain rule on $\sin(xy)$ picks up a factor of $y$, the derivative of $xy$ with respect to $x$):

Differentiating with respect to $y$ instead:

Worked example: mixed second partial derivatives agree

For a well-behaved function, the order of differentiation shouldn't matter: $\dfrac{\partial^2 f}{\partial x\partial y} = \dfrac{\partial^2 f}{\partial y\partial x}$. Check this for the same $f$ above.

Differentiating $\dfrac{\partial f}{\partial x}$ with respect to $y$:

and differentiating $\dfrac{\partial f}{\partial y}$ with respect to $x$:

Both routes land on the exact same expression, confirming the mixed partials agree for this $f$, not just asserting that they should.