Introduction to Partial Derivatives: Fresh Take

  • Calculate the partial derivatives of a function of two variables
  • Calculate the partial derivatives of a function of more than two variables
  • Find higher-order partial derivatives of a function of two variables

Derivatives of a Function of Two Variables

The Main Idea 

With two independent variables, we can’t just take “the” derivative. Instead, we take partial derivatives—one with respect to each variable—by holding the other variable constant.

For [latex]z = f(x, y)[/latex], we have two partial derivatives:

  • [latex]\dfrac{\partial f}{\partial x}[/latex]: measures how [latex]f[/latex] changes as [latex]x[/latex] changes (holding [latex]y[/latex] fixed)
  • [latex]\dfrac{\partial f}{\partial y}[/latex]: measures how [latex]f[/latex] changes as [latex]y[/latex] changes (holding [latex]x[/latex] fixed)

How to Calculate

Treat the other variable as a constant and differentiate normally.

For [latex]\dfrac{\partial f}{\partial x}[/latex]: Treat [latex]y[/latex] as constant, differentiate with respect to [latex]x[/latex]

For [latex]\dfrac{\partial f}{\partial y}[/latex]: Treat [latex]x[/latex] as constant, differentiate with respect to [latex]y[/latex]

Geometric Meaning

  • [latex]\dfrac{\partial f}{\partial x}[/latex]: slope of the tangent line parallel to the [latex]x[/latex]-axis
  • [latex]\dfrac{\partial f}{\partial y}[/latex]: slope of the tangent line parallel to the [latex]y[/latex]-axis

Think of standing on the surface and walking parallel to the [latex]x[/latex]-axis. The steepness you feel is [latex]\dfrac{\partial f}{\partial x}[/latex].

All the Old Rules Apply

Power rule, product rule, quotient rule, chain rule—everything from single-variable calculus works. Just remember which variable you’re differentiating with respect to.

Use the definition of the partial derivative as a limit to calculate [latex]\partial f/\partial x[/latex] and [latex]\partial f/\partial y[/latex] for the function

[latex]f\,(x,\ y)=4x^{2}+2xy-y^{2}+3x-2y+5.[/latex]

Calculate [latex]\partial f/\partial x[/latex] and [latex]\partial f/\partial y[/latex] for the function [latex]f\,(x,\ y)=\tan{(x^{3}-3x^{2}y^{2}+2y^{4})}[/latex] by holding the opposite variable constant, then differentiating.

Watch the following video to see the worked solution to the above example.

Use a contour map to estimate [latex]\partial f/\partial y[/latex] at point [latex](0,\ \sqrt{2})[/latex] for the function

[latex]f\,(x,\ y)=x^{2}-y^{2}[/latex].

Compare this with the exact answer.

Functions of More Than Two Variables

The Main Idea 

Partial derivatives extend naturally to three or more variables. The principle stays the same: hold all other variables constant and differentiate with respect to one variable at a time.

For [latex]w = f(x, y, z)[/latex], we have three partial derivatives:

  • [latex]\dfrac{\partial f}{\partial x}[/latex]: hold [latex]y[/latex] and [latex]z[/latex] constant, differentiate with respect to [latex]x[/latex]
  • [latex]\dfrac{\partial f}{\partial y}[/latex]: hold [latex]x[/latex] and [latex]z[/latex] constant, differentiate with respect to [latex]y[/latex]
  • [latex]\dfrac{\partial f}{\partial z}[/latex]: hold [latex]x[/latex] and [latex]y[/latex] constant, differentiate with respect to [latex]z[/latex]

The number of partial derivatives equals the number of independent variables. Four variables? Four partial derivatives. Five variables? Five partial derivatives.

Problem-Solving Strategy

Identify which variable you’re differentiating with respect to. Treat all other variables as constants. Apply your standard differentiation rules.

Use the limit definition of partial derivatives to calculate [latex]\partial f/\partial x[/latex] for the function

[latex]f\,(x,\ y,\ z)=2x^{2}-4x^{2}+2y^{2}+5xz^{2}-6x+3z-8[/latex].

Then, find [latex]\partial f/\partial y[/latex] and [latex]\partial f/\partial z[/latex] by setting the other two variables constant and differentiating accordingly.

Calculate [latex]\partial f/\partial x,\ \partial f/\partial y[/latex], and [latex]\partial f/\partial z[/latex] for the function [latex]f\,(x,\ y,\ z)=\sec{(x^{2}y)}-\tan{(x^{3}yz^{2}})[/latex].

Watch the following video to see the worked solution to the above example.

Higher-Order Partial Derivatives

The Main Idea 

Just as you can take second, third, and higher derivatives in single-variable calculus, you can take higher-order partial derivatives. With two variables, there are four second-order partial derivatives.

For [latex]f(x, y)[/latex]:

  • [latex]\dfrac{\partial^2 f}{\partial x^2} = f_{xx}[/latex]: differentiate twice with respect to [latex]x[/latex]
  • [latex]\dfrac{\partial^2 f}{\partial y^2} = f_{yy}[/latex]: differentiate twice with respect to [latex]y[/latex]
  • [latex]\dfrac{\partial^2 f}{\partial x \partial y} = f_{xy}[/latex]: differentiate first with respect to [latex]x[/latex], then [latex]y[/latex]
  • [latex]\dfrac{\partial^2 f}{\partial y \partial x} = f_{yx}[/latex]: differentiate first with respect to [latex]y[/latex], then [latex]x[/latex]

The last two are called mixed partial derivatives because they involve differentiating with respect to different variables.

Reading the Notation

  • In [latex]\dfrac{\partial^2 f}{\partial y \partial x}[/latex]: read right to left (differentiate [latex]x[/latex] first, then [latex]y[/latex])
  • In [latex]f_{xy}[/latex]: read left to right (differentiate [latex]x[/latex] first, then [latex]y[/latex])

Clairaut’s Theorem

If the mixed partial derivatives are continuous, then [latex]f_{xy} = f_{yx}[/latex]. The order doesn’t matter.

This holds for most functions you’ll encounter. It means you can choose whichever order makes the calculation easier.

Problem-Solving Strategy

Find the first partial derivatives. Then differentiate each one with respect to both variables. Check that the mixed partials match—they should if the function is well-behaved.

Calculate all four second partial derivatives for the function

[latex]\large{f\,(x,\ y)=\sin{(3x-2y)}+\cos{(x+4y)}.}[/latex]

 

Watch the following video to see the worked solution to the above example.

Partial Differential Equations

The Main Idea 

A partial differential equation (PDE) involves an unknown function of multiple variables and its partial derivatives. These equations model physical phenomena like heat flow, wave propagation, and fluid dynamics.

Common Examples

  • Heat equation: [latex]u_t = c^2(u_{xx} + u_{yy})[/latex] describes temperature distribution over time
  • Wave equation: [latex]u_{tt} = c^2(u_{xx} + u_{yy})[/latex] describes vibrations and wave motion
  • Laplace’s equation: [latex]u_{xx} + u_{yy} = 0[/latex] describes steady-state phenomena

In these equations, [latex]x[/latex] and [latex]y[/latex] represent spatial coordinates, and [latex]t[/latex] represents time.

Verifying Solutions

To verify a proposed solution:

  1. Calculate all required partial derivatives
  2. Substitute them into the PDE
  3. Simplify to check if both sides match

Visualizing Solutions

Solutions to PDEs are functions of multiple variables. For the heat equation with time, we can graph snapshots at different times to see how temperature evolves. As time increases, extreme temperatures level out.

Problem-Solving Strategy

Calculate the needed partial derivatives carefully. Substitute into the PDE. Simplify both sides. Verify equality.

Verify that [latex]u\,(x,\ y,\ t)=2\,\sin\,(\frac{x}{3})\,\sin\,(\frac{y}{4})\,e^{-25t/16}[/latex] is a solution to the heat equation

[latex]u_{t}=9(u_{xx}+u_{yy})[/latex].

Watch the following video to see the worked solution to the above example.