Exponential functions

An exponential function has the form  y = a^x, where a, the base, is a positive number typically greater than 1.  Exponential functions are continuous and defined for all values of x.

Exponential functions exhibit the property of exponential growth.  For we say that a quantity grows "exponentially" when it grows at a rate that is proportional to its size.  That is, the bigger it is at any given time, the faster it's growing at that time.  A standard example is population.  The more individuals there are, the more births there will be, and hence the greater the rate of change of the population -- the number of births per year.

As another example, consider this salary offer:  You are going to be hired for 30 days starting tomorrow, and you are offered two options: 1)  $1,000,000 at the end of the 30 days, or  2)  1 cent now (day 0), and when you start tomorrow (day 1), your pay will double each day.  That is, on the first day you will get 2 cents; on the second day, 4 cents; and so on.

Which option will you take?

If you know anything about the powers of 2 -- which grow exponentially -- you will quickly take the second option.  Here, in fact, is a table of your daily pay:

Such are the rewards of patience.  This is exponential growth.  Your pay grows at a rate -- the number of dollars per day -- that is proportional to the pay itself that day.

Thus on day 0, you're paid 1 cent.  Therefore the rate at which you're paid that day is 1 cent per day (almost) -- so that the next day you earn 2 cents.  But on day 29 you're paid $5,368,709.12.  Therefore the rate at which you're paid on that day is $5,368,709.12 per day (almost) -- so that the next day you earn $10,737,418.24

The exponential function here was  y = .01 × 2^x.   The values of x were the integers 0, 1, 2, 3, etc.

Now, to say that the rate of growth is proportional to its size, is to say that

"The derivative of e^x with respect to x

is equal to e^x."

The derivative of e with a variable exponent

That is,

"The derivative of e with a variable exponent

is equal to e with that exponent
times the derivative of that exponent."

Example 1.   Calculate the derivative of e^{2x + 3}.

Solution.

Problem 1.   Calculate the derivative of e^x².

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e^x²· 2x = 2x e^x²

Problem 2.   Calculate the derivative of the following.

a)   e^{sin x}.    e^{sin x} cos x

b)  e^−x.    e^−x (−1) = −e^−x

The system of natural logarithms

Here are the inverse relations:

And the logarithm of the base itself is always 1:

1.    ln ab  =  ln a  +  ln b.

3.    ln aⁿ  =  n ln a.

The derivative of ln x

We will now prove:

For, since

e^{ln x} = x,

then on taking the derivative of both sides:

equal 1:

	=	eln x·
	=

This implies

Which is what we wanted to prove.

The derivative of the natural logarithm of a function

Example 2.

Example 4.   Find the derivative of ln x².

Solution.   We may apply the laws of logarithms:

The general power rule

We can now prove that for any exponent n:

Let

	y	=	xn.
Then
	ln y	=	n ln x  (3rd Law).
Therefore, on taking the derivative with respect to x:
		=	n x
so that
	y'	=	n x	· y
		=	n x	· xn
		=	nxn−1.

This is what we wanted to prove.

Problem 5.   Calculate the derivative of

The derivative of a^x

We will prove:

"The derivative of an exponential function with base a

is equal to the natural logarithm of the base

times the exponential function."

Let

y	=	ax.
Then on taking the natural logarithm of both sides:
ln y	=	x ln a.  (3rd Law)
Therefore,
	=
But by the chain rule:
	=
Therefore,
			=	ln a
			=	ln a
		y'	=	ln a· y
That is,
			=	ln a· ax.

This is what we wanted to prove.

Problem 6.   Calculate the derivative of y = 10^5x.

By the chain rule:

The constant of proportionality

That limit is the constant of proportionality.  We will therefore define the base of our logarithms to be that number a such that that limit will be 1.

To see that there is such a number a, consider these graphs:

On the left, a = 2, and on the right, a = 3.  (h = 0 is a removable discontinuity, for the vertical line h = 0 is not an asymptote to the graph.)

Clearly, there will be a value of a greater than 2 on the left, and less then 3 on the right, such that the graph passes exactly through y = 1.

That will be the number we call e.