10. Partial Derivatives

by M. Bourne

So far in this chapter we have dealt with functions of single variables only. However, many functions in mathematics involve 2 or more variables. In this section we see how to find derivatives of functions of more than 1 variable.

This section is related to, but is not the same as Implicit Differentiation that we met earlier.

Example - Function of 2 variables

Here is a function of 2 variables, x and y:

F(x,y) = y + 6 sin x + 5y²

To plot such a function we need to use a 3-dimensional co-ordinate system.

Partial Differentiation with respect to x

"Partial derivative with respect to x" means "regard all other letters as constants, and just differentiate the x parts".

In our example (and likewise for every 2-variable function), this means that (in effect) we should turn around our graph and look at it from the far end of the y-axis. So we are looking at the x-z plane only.

We see a sine curve at the bottom and this comes from the 6 sin x part of our functionF(x,y) = y + 6 sin x + 5y². The y parts are regarded as constants.

(The sine curve at the top of the graph is just where the software is cutting off the surface - it could have been made it straight.)

Now for the partial derivative of

F(x,y) = y + 6 sin x + 5y²

with respect to x.

The derivative of the 6 sin x part is 6 cos x. The derivative of the y-parts is zero since they are regarded as constants.

Notice that we use the symbol "∂" to denote "partial differentiation", rather than "d" which we use for normal differentiation.

Partial Differentiation with respect to y

"Partial derivative with respect to y" means "regard all other letters as constants, just differentiate the y parts".

As we did above, we turn around our graph and look at it from the far end of the x-axis. So we see (and consider things from) the y-z plane only.

We see a parabola. This comes from the y² and y terms in F(x,y) = y + 6 sin x + 5y². The "6 sin x" part is now regarded as a constant.

Now for the partial derivative of

F(x,y) = y + 6 sin x + 5y²

with respect to y.

The derivative of the y-parts with respect to y is 1 + 10y. The derivative of the 6 sin x part is zero since it is regarded as a constant when we are differentiating with respect to y.

Second Order Partial Derivatives

We can find 4 different second-order partial derviatives, as follows.

(1)

This could also be written as

This expression means "find the partial derivative with respect to y of the partial derivative with respect to x".

In our example above, we found

To find , we need to find the partial derivative with respect to y of .

Since cos x is a constant (when we are considering differentiation with respect to y), its derivative is just 0.

(2)

This could also be written as

This expression means "find the partial derivative with respect to x of the partial derivative with respect to y".

In our example above, F(x,y) = y + 6 sin x + 5y², we found

To find , we need to find the partial derivative with respect to x of .

Since y is a constant (when we are considering differentiation with respect to x), its derivative is just 0.

(3)

This could also be written as

This expression means "find the partial derivative with respect to x of the partial derivative with respect to x".

In our example above, we found

To find , we need to find the derivative with respect to x of .

(4)

This could also be written as

This expression means "find the partial derivative with respect to y of the partial derivative with respect to y".

In our example above, F(x,y) = y + 6 sin x + 5y², we found

To find , we need to find the derivative with respect to y of .

9. Higher Derivatives

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