The following picture of a white board illustrates step by step how to find the derivative of a function using the definition of the derivative. This process is much harder from an algebraic point of view than using the differentiation rule for x raised to an integer power.
The function we are taking the derivative of is the sum of a number and its reciprocal.
The first step is substituting the function into the limit definition of the derivative. The second step is combining like terms in the numerator of the difference quotient. The third step observes that the limit of a constant is just the constant which is one in this case. The fourth step combines two fractions into a single fraction. The fifth step involves the cancelling of the h in the numerator and denominator. The last step takes the limit by substituting 0 in for h to get a derivative of -1 divided by x squared.