We have seen in the previous page how the derivative is defined: For a function f(x), its derivative at x=a is defined by
Let us give some examples.
Example 1. Let us start with the function f(x) = x2. We have
So
which means f '(a) = 2a.
What about the derivative of f(x) = xn. Similar calculations, using the binomial expansion for (x+y)n (Pascal's Triangle), yield
Example 2. Consider the function f(x)=1/x for . We have
Consequently,
Have you noticed? The algebraic trick in both of the examples above has been to factor out "h" in the numerator, so that we can cancel it with the "h" in the denominator! This is what you try to do whenever you are asked to compute a derivative using the limit definition.
I hope this helps. :)