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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