It is important to know calculus for the further study of just about any subject. Although we could take a statistics course and never seen any calculus at all, it is there, lurking beneath the surface. Further study in statistics, such as a mathematical statistics course, will require all sorts of applications of calculus.
One of the fundamentals of calculus is a process known as differentiation. This is also known as taking the derivative of a function. A few rules go a long way to differentiate a wide variety of functions. One of the most basic of these rules is called the power rule.
Powers and Exponents
In mathematics we have many shorthand notations. Exponents are one of these conventions. If we want to write repeated multiplication of the same number or quantity, we use a number called an exponent or power. The power tells us the number of times that we multiply the base number. So instead of writing 2 x 2 x 2 x 2 we could instead denote this as 24. The four tells us to multiply four copies of two together. Here the four is the exponent and two is the base.
There are a couple of additions to this definition that are necessary to treat other sets of possible exponents. One of these is that a fractional exponent corresponds to a root of the number. So 1001/2 is the square root of 100 and 641/3 is the cube root of 64.
A second addition to the definition is that a negative exponent tells us to first take the reciprocal of the base before applying the exponent. So 5-1 = 1/5 and 4-2 = 1/16.
A Statement of the Power Rule
Now that we have carefully considered exponents, we are in a position to talk about the power rule. The power rule says that for any real number n, the derivative of xn is the function nxn – 1. We can write the derivative of the function f( x ) with a prime, f’( x ). And so the power rule is stated as
( xn)’ = nxn – 1.
One example of the power rule at work is in finding the derivative of the function f( x ) = x5. A straightforward application of the rule shows that f’( x ) = 5x4.
For another example of the power rule we see that if g( x ) = x then g’(x) = 1x0 = 1.
Derivatives of Constants
The derivative of any constant function is equal to zero. This fact is sometimes referred to as the constant rule. This rule is really a consequence of the power rule for the case when n = 0.
Since x0 = 1, any constant c can be expressed as cx0. We use the power rule and see that
(c)’ = (cx0)’ = 0cx-1 = 0.
Derivatives of Polynomials
Although the power n in the power rule can be any real number, by restricting these powers to be nonnegative whole numbers we can easily find the derivative of any polynomial. To do this we need to recall that the derivative is a linear operator. This means that we can factor out constant terms and that the derivative of a sum of functions is the sum of the derivative of those functions.
Let p( x ) = cnxn + cn - 1xn - 1 + . . . + c2x2 + c1x + c0. Here every ci is a real number and n is a whole number.
The derivative is calculated to be
p’( x ) = ncnxn - 1 + (n – 1)cn - 1xn - 2 + . . . + 2c2x + c1.
By this formula, the derivative of h( x ) = 5x3 – 4x2 +6x – 9 is found to be h’( x ) = 15x2 - 8x + 6
Applications of the Power Rule
In statistics the power rule shows up in a few places. Since this is a part of calculus, anywhere that we need to calculate derivatives in mathematical statistics we will run into the power rule. The derivative of a cumulative distribution function is a probability distribution function. Many times the power rule shows up in this context.