Unit 13.1–13.3: Functions of Several Variables & Partial Derivatives

Function of two variables

z=f(x,y)

Graph of multivariable function

Surface in 3D space.

Level curve

f(x,y)=k

Purpose of level curves

Contour map of surface.

Limit of multivariable function

Must approach same value from every path.

Path test

If two paths give different answers, limit DNE.

Partial derivative with respect to x

Hold y constant.

Partial derivative with respect to y

Hold x constant.

Notation

fₓ=∂f/∂x

Second partial derivative

fₓₓ=∂²f/∂x²

Mixed partial derivative

fₓᵧ=∂²f/(∂y∂x)

Clairaut's theorem

If partials are continuous, fₓᵧ=fᵧₓ.

How to compute fₓ

Pretend y is a constant.

How to compute fᵧ

Pretend x is a constant.

How to show limit DNE

Try y=0 and x=0 first.

Professor trick

Always test y=mx if simple paths agree.

Common mistake

Differentiating both variables at once.