C. Function of a Function

Suppose we want to differentiate (2x -1)³. We could expand the bracket then differentiate term by term, but this is tedious! We need a more direct method for expressions of this kind.

Now (2x-1)³ is a cubic function of the linear function (2x -1), i.e. it is a function of a function.

Other examples

(x² - 3)³ is a cubic function of a quadratic function

is a square root function of a quartic function

There are 2 ways to think about solving functions of a function:

(i) Chain Rule

If we have y(x) = f (complicated expression), we let u = (complicated expression) then work out and . We then use:

      The Chain Rule

Examples

1. y = (2 - x³)⁴

let u = 2 - x³, so that y = u⁴

= 4u³ and = -3x²

So = (4u³).(-3x²) = -12x²(2 - x³)³

2. y(x) = , i.e. y = (1 - x²)^-1

let u = (1 - x²), so that = -2x

and y = u^-1, so that = -

= (-2x)(-) = +

(ii) Sequential Step Method

With this method, we start with the outermost function, and differentiate our way to the centre, multiplying everything together along the way.

Examples

1. y = (2 - x³)⁴

think of this as y = (expression)⁴

differentiating, dy/dx = 4(expression)³

We now look at the expression in the brackets and differentiate that (= -3x²) and multiply it to our previous answer to give

= 4(2 - x³)³(-3x²) (which is the same as before)

2. y = = (1 - x²)^-1

= -(1 - x²)^-2 . (-2x)

3. y = = (x² - 1)^½

= ½(x² - 1)^-½ . 2x

4. Refer back to this later, after we've covered sin and ln.

y = sin{ln(3x² + 2)}

= cos{ln(3x² + 2)} . . 6x

The general expression for an exponential function is

An example is y = 3^x

x	0	1	2	3	4	etc
y	1	3	9	27	81	etc

One of the most important properties of an exponential function is that the slope of the function at any value is proportional to the value of the function itself.

In other words, proportional to y(x), or = constant y(x)

the value of the constant depends upon the function y(x).

Numerical examples

1. y = 2^x, plot the graph and measure the slopes at different values of x.

x	y	slope at x measured from graph	slope / y
0	1	0.69	0.69
1	2	1.38	0.69
2	4	2.76	0.69
3	8	5.52	0.69
4	16	11.04	0.69

So for y = 2^x, the constant is 0.69 (later on we'll see this is ln 2).

x	y	slope	slope / y
0	1	1.1	1.1
1	3	3.3	1.1
2	9	9.9	1.1
3	27	29.7	1.1
4	81	89.0	1.1

So for y = 3^x, the constant = 1.1

Now, in the above 2 examples we used simple numbers (a = 2 and a = 3), but the constants were not simple numbers.

But we can reason that there must be some number between 2 and 3 for which the constant = 1, exactly.

i.e = y(x)

The value of a that gives this result is known as "e" and has the value:

an irrational number.

e is actually calculated from the following progression formula:

e =

The Exponential Function

The function e^x is known as the exponential function (as opposed to any other exponential function) and is extremely important in all branches of science:

• Radioactive materials undergo exponential decay,
• World human population is increasing exponentially,
• Chemical reaction rates depend exponentially upon the temperature, etc.

What does e^x look like?

x	0	1	2	3	4	5
ex	1	2.72	7.39	20.1	54.6	148
e-x	1	0.37	0.14	0.05	0.02	0.007