Partial Derivatives

Abstract

You take derivatives as usual, holding the other variable constant.

On a surface , we have many derivatives possible, for example,

derivative with respect to .

Definition

which is similar to the definition of a normal derivative.

Example

Find the partial derivatives of:

solution

Geometrically, this can interpreted as a slice, and the slope of the tangent along x or y.

Remark

All other differentiation methods apply here, such as implicit differentiation

Example

Find the slope of a tangent line to the surface in the y direction tangent to
solution

If we re-write the function as and , we find that this is a sphere with radius 1.

We want ant with
Instead, we'll use implicit differentiation (easier)

Remark

Observe that the coordinate doesn't matter. Look at the figure to see why this is intuitively and geometrically the case.

Higher Order Partial Derivatives

Example

Find the second derivatives of:

solution

Clairaut's Theorem

Theorem

If the second partial derivatives of a function are continuous, then the order of differentiation is immaterial.

Let have all partial derivatives up to second order continuous near . Then .

Proof

By definition:

Applying the mean value theorem to the function on , we find that there is a value such that:

which gives

Since as (by definition of ), and because is continuous, we have:

which is the desired result.

Example

Using the same above example, would be differentiating with respect to :

If we find , it's the same!

Remark

For the vast majority of functions, the order of partial derivatives will not matter (i.e they are commutative). Functions who's partial derivatives do not commute have no practical purpose.

You'd have to me looking for them on purpose. Profs like to look for them to torture math students. For our purposes we won't worry about it.
- Prof. Matthew Scott