Lesson Plans
Lesson 5
Operations on Derivatives; Derivatives
of Polynomials
By
Michael McCallum
Introduction:
In this lesson we will show how to calculate the derivative of a constant and a monomial and compare the derivative to the original function. We will use this comparison and the fact that the derivative is a linear operator to show how to calculate the derivative of a polynomial without using the difference quotient.
The Derivative of a Constant
First, let us calculate the derivative of a constant function. Let f (x) = 5. Using the definition of the derivative we have:
So the derivative of a constant is zero.
The Derivative of a Monomial
Let f (x) = x3, and calculate the derivative using the definition.
Now let f (x) = x2, and calculate the derivative using the definition.
There seems to be a pattern here. The derivative of x3 is 3x2, and the derivative of x2 is 2x. In fact, the derivative of a general monomial xn is nxn-1.
Operations on Derivatives
A derivative is a function derived from another function. As such, it has the same properties as any function. Namely, we can perform the same operations on a derivative as we can on any other function. That is, if we had the function f(x) = x3 + x2, we could calculate the derivatives of the terms of the function individually and then add them to determine the derivative of f. In this case dy/dx = 3x2 +2x. If the coefficients of the terms in the function were other than 1, we can multiply the result of calculating the derivative as if its coefficient were 1 and then multiplying the result by the actual coefficient. This greatly simplifies the calculations of derivatives. It is not really necessary to use the difference quotient definition for the derivative every time you need to calculate a derivative.
Derivatives of Polynomials
We can take advantage of the discoveries in the previous paragraph when calculating the derivative of a polynomial. We simply calculate the derivative of each term in the polynomial separately as if it were a monomial and then combine these individual derivatives using the original operators from the polynomial. For example, let f(x) = 5x6 –3x4 +2x3 –8x2 + 4x –7. Calculating the individual derivatives of the terms and recombining them we get: dy/dx =30x5 –12x3 + 6x2 –16x + 4. Here we notice that the degree of the derivative is one less than the degree of the function. This will always be true. Calculate several more derivatives of polynomials and have the students calculate several at the board or at their desks to conclude this lesson.