To find the derivative of the logarithm with base b, we began by writing y=log[b](t) in the exponential form t=b^y. Now, using the above rule, we can find dt/dy, and then take reciprocals to get dy/dt. We did this, and found that dy/dt=1/(t*ln(b)). As a special case worth mentioning, if b=e, then we have the derivative of ln(t) is 1/t. This is the last of our rules for derivatives. We did a few more practice problems in class finding derivatives by hand, this time incorporating exponential and logarithmic functions into the mix.
Finally, we took a look at functions defined by a table of values. Since we only know values at certain times, we cannot take limits, so we have to settle for a "reasonable" difference quotient. we saw three ways to do this: forward difference, backward difference, and symmetric difference (which is actually the average of the forward and backward differences). The forward difference is just the difference quotient with h>0, taking h as small as the data will allow. The backward difference is the similar quotient with h<0. In practice, the symmetric difference often yields a better approximation.
Now, at this point you have all the
techniques for finding derivatives. The gateway test will be available
starting June 22. Remember, you can take it multiple times, but it a requirement
for completion of this course that you pass with 90% proficiency. The test
consists of 10 functions for you to differentiate. Simplification of answers
is not important, but accuracy is--there is no partial credit. You must
get 9 out of 10 right to pass. I gave out sheets of practice problems in
class for you to work with to refine your skills. I will be available to
help you with difficulties, but this is something you are expected to do
on your own--we will not spend more time in class on techniques for finding
derivatives.
