Other Bases

Exponentials With Other Bases

 Definition   Let a > 0 then           ax = ex ln a

Examples

Find the derivative of the following functions

1. f(x) = 2x

2. f(x) = 3sin x

3. f(x) = xx

Solution

1. We write

2x = ex ln 2

Now use the chain rule

f '(x) = (ex ln 2)(ln 2) = 2x ln 2

2. We Write

3sin x = e(sin x)(ln 3)

Now use the chain rule

f '(x) = (e(sin x)(ln 3))(cos x)(ln 3)

= (3sin x) (cos x) (ln 3)

3. We Write

xx = ex ln x

Notice that the product rule gives

(x ln x)' = 1 + ln x

So using the chain rule we get

f '(x) = ex ln x (1 + ln x) = xx (1 + ln x)

Exercises

Find the derivatives of

1.  x2x + 1

2. x4

Logs With Other Bases

 Definition                           ln x      loga x  =                                      ln a

Examples

Find the derivative of the following functions

1. f(x) = log4 x

2. f(x) = log (3x + 4)

3. f(x) = x log(2x

Solution

1. We use the formula

f(x) = ln x / ln 4

so that

f '(x) = 1/(x ln 4)

2. We again use the formula

ln (3x + 4)
f (u)  =
ln 10

now use the chain rule to get

3 ln (3x + 4)
f '(x)  =
ln 10

3. Use the product rule to get

f '(x) = log(2x) + x(log 2x)'

Now use the formula to get

ln 2x          x ( ln 2x)'
f ' (2x)  =                   +
ln 10             ln 10

The chain rule gives

ln 2x                  2x
f ' (x)  =                        +
ln 10               2x ln 10

ln 2x                 1
f ' (x)  =                        +
ln 10               ln 10

Integration

Example

Find the integral of the following function

f(x) = 2x

Solution

2x dxex ln 2 dx         u = x ln 2,    du = ln 2dx

1
=               eudu
ln 2

1                               ex ln 2
=                 eu  +  C  =
ln 2                             ln 2

2x
=                +  C
ln 2

Application:  Compound Interest

Recall that the interest formula is given by:

A = P(1 + r/n)n

where n is the number of total compounds before we take the money out, r is the interest rate,
P is the Principal and A is the amount the account is worth at the end.
If we consider continuous compounding, we take the limit as n approaches infinity we arrive at

A = Pert

Exercise

Students are given an exam and retake the exam later.  The average score on the exam is

S = 80 - 14ln(t + 1)

where t is the number of months after the exam that the student retook the exam.  At what rate is the average student forgetting the information after 6 months?