1. The concept of the derivative of the function is exceptionally important. The derivative shows that how fast the function changes when its argument is changed. (Remember that for f ( x) we say that f is a function that depends upon the argument x. You should think of f as a machine which gives you the value f when you input x.)
2. Functions do not always have to be written as f ( x). us where a body is at different times t.
3. The derivative of x(t ) at time t can be defined as:
4. Now let's see how to calculate the derivative of a simple function such as x(t ) = t 2 . We should first calculate the difference in x at two slightly different values say, t and t + Δt, while that recall we choose Δt to be extremely small:
5. In exactly the same manner you can show that if x(t ) = t n then:
This is an exceptionally useful result.