1. No warm up today.

2. 5.5 Bases other than e
Objective: To take the derivative and integrals of other exponential functions.
Questions?
y = ax  has a constant base and variable exponent.
YOU CANNOT USE THE POWER RULE! WHY?
You must use logarithmic differentiation or memorize a
formula.
x
3x
Ex1. y = 2
Ex2: y = 5
Do you notice a pattern with the answers???????

3. If y = au, then y’ = (au)(lna)(u’)
How is this the same for y = eu?
Try these:
1. y
6
3x
2. y
4
x3
But WAIT!! This short-cut will only work if
the base is a constant. 
Remember: If y = variable base variable exponent  you MUST use
log differentiation. The following have NO shortcut.
1) y = xsinx
2) y
x 2x

4. Derivative of y = log a x
u 1
If y = log a u, then y’ =
u ln a
Example:
Find y’ for each of the following.
1. y = log5 x2
2. y = log3 7x

5. Integrals of bases other than e.
1
x
a dx
a C
(ln a)
x
1. 2 x dx
Be aware, you may have to use u-substitution!!
2. 25 x dx

6. Examples:
1. x 5
x2 2
dx
43 x
3.
dx
3x
1 4
2.
sin x
cos xdx