### Example 1

Find and

So therefore:

Now,

Note:

### Example 2

Find

We have,

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## MA101.21 Examples of the Chain Rule and Partial Differentiation

### Example 1

### Example 2

## MA101.20 Generalisation of the chain rule to partials

###### Special cases

## MA101.19 The Chain Rule – One Variable

## MA101.18 Partial Differentiation

##### Examples

## MA101.17 Continuity of functions of two variables

##### Example

## MA101.16 Functions of two variables or more

##### Example

##### Example

##### Example

##### Example

## MA101 Assignment 2 – Limits and Continuity

## MA101.15 A bit more on limits

##### Examples

## MA101.14 The hyperbolic functions

##### Derivative of sinh and cosh

##### Inverse functions

##### Derivatives of the inverse hyperbolic functions

## MA101.13 Trigonometrical functions

##### Theorem

###### Proof

Find and

So therefore:

Now,

Note:

Find

We have,

Consider the following simple example. Let f be a function of two variables defined by:

where .

By substituting,

Let us define a function F of two variables by

Then we can calculate

With the obvious intentions regarding the partial derivatives we can also calculate

It can also be checked that

These two results are indeed true for a general function and substitution . They are regarded as a generalisation of the chain rule for one variable. Again and are often confusingly written as and .

The rule is then

It is very important to remember what is meant by all these items.

Note the chain rule can be used both ways.

ie let , , define

We have

But also defines (substituting for u and v in terms of x and y)

So,

and, of course, we put F as f throughout.

These results are special cases of the so called general chain rule.

If we have a function where x are substitutions

Then

i)

ii)

Consider the function given by . We are used to wring such things as:

i)

ii)

iii) . For example we would write for example.

Equally well, of course, it would be true to write .

The meaning of (i) and (ii) are mathematically precise. means the derived function and means the value of at x.

The meaning of can be more devious.

It can simply be taken as synonymous with . That is .

When such is the intention it would be indisputable that means .

But there are other more shady uses as we will see.

Now consider substituting in to define a function F defined as .

The chain rule says

where here means .

Note that F is not equal to f, but mathematicians frequently write the chain rule as,

.

Here does not mean which is after all .

To see the chain rule in a more precise and unambiguous form think of as defining a function g given by , then and we see the chain rule as saying

Of course the u here is an entirely dummy symbol.

Consider the curve in which the surface meets the plane a constant.

In this plane , and would be a formula for the gradient of the tangent to the curve.

If we differentiate with respect to x, treating the y as if it was a constant (some say holding y constant), the the derivative obtained is called the partial derivative of with respect to x and we write or or . Similarily we have .

With functions of more than two variables one differentiates with respect to one of the variables by holding all the other variables constant.

.

In the obvious way we can have higher order partial derivatives.

Similarly for we have:

For commonly encountered functions we have that

From here on we may assume that all the mixed derivatives are equal.

Note, the normal rules (sum, product, quotient, function of a function) of differentiation apply to partial differentiation.

We can say that as the , or .

If, intuitively speaking, by going close enough to (a,b) we can get f(a,b) as close as we like to L.

f(x,y) is said to be continuous at (a,b) if .

Just as with one variable limits from below and above may be different, with two variables we may get various limits coming in from different paths.

Consider coming in to (0,0) along the line y=mx (m fixed). Along this line we have:

Thus along y=mx,

If we come in along the line ,

.

For a function of two (real) variables each element of its domain is an ordered pair of real numbers (x,y) only, and each element of its range is a real number, z, say.

x and y are called independent variables, and z the dependent variable.

To specify such functions we must have both a rule, and a domain.

i)

ii)

If no known domain is specified we again take it to be the largest subset of for which the rule makes sense.

A graph of may be drawn in the usual way. (ie. (x,y) in the horizontal plane called the xy plane, and z moving along a third, vertical axis.

For obvious reasons informative graphs are often quite difficult to draw. It’s often useful to draw profiles of f, that is the surface curves where their surface meets planes parallel to the coordinate axis.

Sketch

Consider the intersection with x=constant, y=constant and z=constant.

Another useful diagram of f(x,y) is that provided by drawing the contour lines (or level curves). That is we sketch in the x,y plane the graph of y=f(x,y) for various values of z.

Look at again.

so

When,

Sketch

For a function we can not sketch profile, and the best we can do is sketch level surfaces.

For etc…

For

for any possible integer m we care to choose.

Choose then .

But as .

So, as .

So it can be said that increases more rapidly than any power of x.

Also (for ).

Where as , and as as .

So it can be said that increases more slowly than any positive power of x.

Also (for ).

Where .

.

We define:

Note that:

– so an odd function.

– so an even function.

Also: as .

Note:

Exercise

Find derivatives of tanh, coth, sech,cosech.

sinh is an injection and so we have an inverse function denoted by .

Domain = range of

Range = domain of

means the real number whose is x.

tanh is an injection and so we have an inverse function denoted by .

Domain = range of

Range = domain of

means the real number whose is x.

Note that is not an injection, so we need a cut down domain to non-negative x’s () to make it one.

means the non-negative real number whose is x.

.

Series definitions for the sin and cosine functions are:

These converge .

If we differentiate these term by term we can see that:

Many other properties can be deduced from these power series.

The graph of is as shown:

We can see that sin is not an injection (domain of ) and so there is no inverse. However the function , or [called the cut down sine], has the graph:

and this is an injection, and has an inverse function with domain $latex[-1,1]$ and range . is the unique real number (angle) between whose sine is x. The is symbolised by or .

The graph of is a reflection of (cut down) in the line .

Knowing is true, you may deduce is true.

Example: .

Knowing is true, you may not deduce .

Example: .

We take the positive square root because for the

Similarly we define the ‘so called’ cut down cosine – this is cosine but with the domain , and the cut down tangent with domain . The inverses of these functions are arccos and arctan.