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.

The general procedure for finding the derivative of a function raised to the fourth power is to bring the 4 to the front and then subtract one from the exponent to get three. Then you multiply by the derivative of original function.

This process is called the Chain rule or specifically the power rule. This process applies for any power not just four.