Variants of the Exponential Function

In most applications the exponent is not simply x, but some function of x, or the exponent is multiplied by some other function of x.

Notation: We normally write the exponential function as y = e^x, but if the exponent is a function (e.g. x² + 1), it's often easier to write it as y = exp(x² + 1), which is equivalent to . (Note - This exp is not the same EXP on your calculator, which refers to 10^x)

Examples
y = e^bx,      y = e^-bx,      y = ,

y = e      or      y = exp(2x² - 1)

y = 5e^x,      y = 5x.e^-bx,      y = (x² - 1)e,      etc.

The reason why e^x is so important is that it is the only function which doesn't change when you differentiate it.

i.e. if       y = e^x,           = e^x

Examples

1. y = 5e^x, = 5e^x

2. y = 3e^x + 2, = 3e^x

3. y = e^bx,   use chain rule with u = bx, = b

and y = e^u, = e^u

So = . = be^bx

Alternatively, using the sequential rule:

= e^bx . b

So a general result is

= be^bx

5. y = exp(x² + 1), = exp(x² + 1) . 2x

6. y = x.e^-bx. This requires the product rule

= u + v = x(-be^-bx) + e^-bx.(1) = (1 - bx)e^-bx

7. What is the gradient of the curve y = 5e^-2x + 1 at the point x = 2?

= -10e^-2x, and at x = 2, slope = -0.183.

6. Logarithmic Functions

If we have the relationship

then there must be the inverse relationship such that

We call the function, f (y), the logarithm to base a.

There are 2 types of logarithm in common use:

a) Common logs have base 10 and are written log₁₀x

b) Natural logs have base e and are written log_ex or ln x

So, if

y = e^x, then ln y = x

y = 10^x, then log₁₀ y = x

Laws of Logarithms

Examples

1. ln 2 + ln 3 = 1.792 = ln 6

2. ln x + ln (x²+1) = ln {x.(x²+1)} = ln (x³+x)

3. ln 6 - ln 3 = 0.693 = ln 2

4. ln (x+1) - ln (3-x²) = ln

5. ln x + ln (x+3) - ln (x²+4) = ln

6. ln (x² + 1)³ = 3ln (x²+ 1)

7. ln (x²+3)^x+1 = (x+1) ln (x²+3)

8. ln = ln(x^-1) = -ln x, (important)

Numerical Example: when x =2,

9. ln = ln = ln = 0.118

or = 2ln (x+1) - ln (4x) = 2ln 3 - ln 8 = 0.118

The Differential of ln x

It can be shown that, if

y = ln x,          =

We can now use this, together with the Product, Chain and Sequential Rules to find the Differentials of log functions.

Examples

1. y = ln (ax + b)

= a =

2. y = ln (2) + 3x² = ln (2x^½) + 3x²

= =

3. What are the stationary points in y = ln(x) - x?

=

So the slope = 0 when = 0, i.e., when x =1, and y = -1.

= -, which at (1,-1) is -ve, so it is a maximum.