Post #17

So after 11 make-up posts I ran out of stuff to say, so i'm going back to the beginning of the year and talking about some stuff.

Complex Derivatives y=ln(e^x) (Chain Rule)

First off one should should identify the steps of your problem. In this case they would be:

1. Natural Log
2. e^x

you problem should be (1/(e^x)).(e^x)'

then you find the derivative of e^x which is e^x . x' (x'=1)

so your final problem should be (1/(e^x)).(e^x)

After this you have to simplify algebraically, giving you (e^x)/(e^x) ,which equals 1.


First Derivative Test:

1. Take the derivative of the original problem.
2. Set the first derivative equal to Zero.
3. Solve for x.
4. Create intervals for x. i.e. (-∞, 1) (1, 4) (4, ∞)
5. Pick a number in the intervals then plug that number in the first derivative for x.
6. Solve. For positive numbers, the graph of the derivative is above the x-axis. For negative numbers, the graph of the derivative is below the x-axis. The numbers for x are your points of inflection. (Points of Inflection are only if there is a shift in the graph!!!)

Second Derivative Test:

1. Take the derivative of the first derivative.
2. Set the second derivative equal to Zero.
3. Solve for x.
4. Create intervals for x. i.e. (-∞, 1) (1, 4) (4, ∞)
5. Pick a number in the intervals then plug that number in the second derivative for x.
6. Solve. For positive numbers, the graph of the derivative is above the x-axis. For negative numbers, the graph of the derivative is below the x-axis. The numbers for x are your points of inflection. (Points of Inflection are only if there is a shift in the graph!!!)