Basic Derivative Rules
Previously we have discussed how a derivative is found using the limit, why the derivative is useful, and how to find it. Now to attempt to simplify or shortcut the derivative finding process, we will look at some general rules that can be used instead of going through the entire limit step-by-step every time.
The first derivative we will look at is the derivative of a linear function in slope-intercept form.
Where m is the slope and b is the y-intercept. Using the limit definition for the derivative we have:
Therefore the derivative of any linear function is, as we would expect, equal to the slope of that line. The next derivative we will look at is a special case of the linear function. If the slope is zero then the linear function is a constant function.
As above, the derivative of a linear function is its slope therefore the derivative of a constant function is also equal to the slope which is zero.
The next rule we can look at is a single independent variable raised to any power. Hence this rule is often called the Power Rule.
Applying the limit process we obtain:
Notice how we used the Binomial Theorem to expand the first term in the numerator. At this point I want us to notice a certain pattern in the coefficients that can be seen in Pascal’s Triangle. The first coefficient will always be 1 (c1=1). This can be seen by looking at the left edge of the triangle. Every first number is 1. Secondly if you look at the diagonal row parallel to, directly adjacent to, and to the right of this first row of 1’s, you notice that the numbers count sequentially as you descend the horizontal rows of the triangle: 1, 2,3,4,5,..etc. These numbers correspond to the second coefficient in the binomial expansion. If we label the horizontal row of the very first 1 at the top of Pascal’s Triangle zero and then incrementally as we go down the triangle, the second row will provide the coefficients for the binomial squared, the third row for the binomial cubed, the forth row for the binomial to the fourth power, and so on such that the nth row gives the coefficients for the binomial to the nth power and the second coefficient will be n also (c2=n). From here we have:
So you can see that the derivative of any variable to an exponent is equal to that same variable times the original exponent to the power of one less than the original exponent. For example:
In summary we can use the following rules to make our lives easier with regard to finding derivatives.