Obligatory application/interpretation problem : Next, we need to do our obligatory application/interpretation problem so we don't forget about them.

Example: Assume that the position of an object is given by  s (t ) = te^t

Does the object stop moving ever?

Solution : First we will require the derivative. We require this to find out if the object ever stops moving as at that point (provided there is one) the velocity is going to zero and recall that the derivative of the position functions is the velocity of the object.

The derivative is,                            s′ (t ) = e^t  + te^t  = (1 + t ) e^t

Hence, we have to determine if the derivative is ever zero. To do this we will have to solve,

                                                                    (1 + t ) e^t  = 0

Now, we already know that exponential functions are never zero and hence this will only be zero at t = -1 . Thus, if we will allow negative values of t then the object will stop moving once at t = -1 .

If we aren't going to let negative values of t then the object will never stop moving.

We should look at couple of derivatives to make sure that we don't confuse the two. The two derivatives are,

d ( xⁿ )/dx = nx ^{n -1}                           Power Rule

d (a ^x )/ dx = a ^x ln a                          Derivative of an exponential function

This is important to note that with the Power rule the exponent should be a constant and the base should be a variable whereas we require exactly the opposite for the derivative of an exponential function.  For exponential function the exponent should be a variable and the base should be a constant.