No warm up today.
5.5 Bases other than e
Objective: To take the derivative and integrals of other exponential functions.

Questions?

y = ax...
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 s...
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 x...
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 ...
Examples:

1. x 5

x2 2

dx

43 x
3.
dx
3x
1 4

2.

sin x

cos xdx
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Bases other than e ap calc notes

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Bases other than e ap calc notes

  1. 1. No warm up today.
  2. 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. 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. 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. 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. 6. Examples: 1. x 5 x2 2 dx 43 x 3. dx 3x 1 4 2. sin x cos xdx

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