MATH 300 Lecture Notes Wenesay, September 25, 203. Section 3. of HH - Powers an Polynomials In this section 3., you are given several ifferentiation rules that, taken altogether, allow you to quickly an easily ifferentiate all power functions an all polynomial functions. Recall that power functions have the form f(x) = kx p, where k an p are any constants. Examples of power functions are f(x) = 2x 3, g(x) = 3x 2, an h(x) = 2x 2 3 Polynomial functions are sums an ifferences of power functions in which the power p is restricte to be non-negative integers 0,, 2, 3, etc.. An example of a polynomial function is p(x) = 2x 3 5x 2 + 6x Now here is an itemize list of the promise ifferentiation rules that, taken altogether, allow you to quickly an easily ifferentiate all power functions an all polynomial functions. (a) The Derivative of a Constant Multiple of a function When you ifferentiate cf(x) with c being any constant, the c can come out in front of the result of ifferentiating the f(x). In symbols: x [c f(x)] = c f (x) This is Theorem 3. on page 6 in your textbook. The proof of this theorem using the limit of a ifference quotient immeiately follows the statement of the theorem on page 6. In wors, we say that multiplicative constants can come out in front of erivatives. See section 2(a) in the Examples-section below for some examples of the use of this rule. (b) The Derivative of a Sum an Difference of functions When you ifferentiate f(x) + g(x), you can ifferentiate each of f(x) an g(x) separately an then a the erivatives together. Likewise, when you ifferentiate f(x) g(x), you can ifferentiate each of f(x) an g(x) separately an then subtract the erivatives. In symbols: x [f(x) + g(x)] = f (x) + g (x) In wors, the erivative of a sum is the sum of the erivatives. x [f(x) g(x)] = f (x) g (x) In wors, the erivative of a ifference is the ifference of the erivatives. Sometimes you will see both of these ifferentiation rules written as one:
x [f(x) ± g(x)] = f (x) ± g (x) where you use the top sign on both sies or the bottom sign on both sies. This is Theorem 3.2 on page 7 in your textbook. The proof of this theorem using the limit of a ifference quotient immeiately follows the statement of the theorem on page 7. See section 2(b) in the Examples-section below for some examples of the use of these rules. (c) The Power Rule The Power Rule allows you to ifferentiate any power function. In symbols: x [xn ] = nx n where n is any constant. The Power Rule is state at the top of page 8 in your textbook, an the justification for the Power Rule for positive integers n is shown at the top of page 9. The justification for the Power Rule for other kins of constants n like negative integer constants, fractions, an inee all real number constants will occur later in the course. Watch for these justifications as the course progresses. Three examples of the use of the Power Rule will be given here, for use in the early parts of the Examples section below. More examples of the Power Rule will be given in the Power Rule section in the Examples below. With n = 2, x [x2 ] = 2x 2 = 2x = 2x With n = 3, x [x3 ] = 3x 3 = 3x 2 With n = 4, x [x4 ] = 4x 4 = 4x 3 See Examples an 2 on page 8 in your textbook for examples of the use of the Power Rule. Also see section 2(c) in the Examples-section below for more examples of the use of the Power Rule. () Polynomials We now have enough ifferentiation rules to be able to ifferentiate any polynomial function. Recall that a polynomial function is a sum an/or ifference of certain power functions. We can now pull apart the sums an ifferences (using The Derivative of a Sum an Difference of functions) an ifferentiate the resulting iniviual terms term-by-term, then pull off any constant in front of each power function term (using The Derivative of a Constant Multiple of a function), an then ifferentiate the bare-bones power function using the Power Rule. A completely worke-out example of the ifferentiation of a polynomial function occurs in the Examples section below in 2(). 2
(e) The Secon Derivative You learne about the secon erivative back in section 2.5 in your textbook. Now we have the tools to be able to compute the secon erivatives of power functions an polynomial functions easily an quickly, because the erivative of a power function is a power function (an so can be easily ifferentiate a secon time), an the erivative of a polynomial function is a polynomial function (an so can be easily ifferentiate a secon time). But the interpretation of the sign of the secon erivative continues to hol: the graph of a function y = f(x) is concave-up (CU) on any interval over which f (x) > 0 for all x in that interval, an the graph of a function y = f(x) is concave-own (CD) on any interval over which f (x) < 0 for all x in that interval. 2. Examples (a) The Derivative of a Constant Multiple of a function The first example here uses x [x2 ] = 2x which was emonstrate above in the section on the Power Rule. x [3x2 ] = 3 x [x2 ] = 3(2x) = 6x The secon example here uses the section on the Power Rule. x [ 7x3 ] = 7 x [x3 ] = 7(3x 2 ) = 2x 2 x [x3 ] = 3x 2 which was emonstrate above in In wors, we say that multiplicative constants can come out in front of erivatives. (b) The Derivative of a Sum an Difference of functions The examples here use the ifferentiation rules emonstrate above in the section on the Power Rule. The secon example here also uses the ifferentiation rule that allows us to bring multiplicative constants out in front of erivatives. x [x4 + x 2 ] = x [x4 ] + x [x2 ] = 4x 3 + 2x x [9x3 5x 2 + 6x 4 ] = x [9x3 ] x [5x2 ] + x [6x4 ] = 9 x [x3 ] 5 x [x2 ] + 6 x [x4 ] = 9(3x 2 ) 5(2x) + 6(4x 3 ) = 27x 2 0x + 24x 3 3
This last example shows that the rules for the erivatives of sums an ifferences of functions are not limite to just sums an/or ifferences of two functions but rather sums an/or ifferences of any number of functions. In wors, we say that the erivative of a sum is the sum of the erivatives. An, the erivative of a ifference is the ifference of the erivatives. (c) The Power Rule With n = 00, x [x00 ] = 00x 00 = 00x 99 With n =, x [x] = x [x ] = x = x 0 = With n = 0, x [] = x [x0 ] = 0x 0 = 0 [ ] With n =, = x x x [x ] = x = x 2 = x 2 With n = 4 5, x [ 5 x 4 ] = x [x 4 5 ] = 4 5 x 4 5 = 4 5 x 5 = 4 5 x 5 = 4 5 5 x = 4 5 5 x Look carefully at the three examples given in Example on page 8 in your textbook. That Example parts (b) an (c) give you more examples about how to use the Power Rule when the constant exponent n is a fraction, either positive or negative. () Polynomials Here s a completely worke-out example, which you shoul strive to eventually be able to accomplish in one step. x [5x3 8x 2 + 3x 6] = x [5x3 ] x [8x2 ] + x [3x] x [6] = 5 x [x3 ] 8 x [x2 ]+3 x [x] 0 [Derivative of sum an ifference] [Multiplicative constants come out in front of erivatives, an for the last term: the erivative of any constant is 0] = 5(3x 2 ) 8(2x) + 3() [Power Rule, applie three times] = 5x 2 6x + 3 You can fin four more completely worke-out examples in your textbook: see Example 3 at the bottom of page 9, an Example 4 near the top of page 20. 4
(e) The Secon Derivative Example 5 starting in the mile of page 20 in your textbook shows three examples of computing secon erivatives an then interpreting their signs to give information about the concavity of the original function. Example 6 starting at the very bottom of page 20 ifferentiates a istance function twice, the first erivative yieling the velocity function an the secon erivative yieling the acceleration function. The first erivative of a istance function is a velocity function because velocity is the rate at which istance changes in time. In turn, the rate at which the velocity changes in time is what we call the acceleration. So the acceleration function is the secon erivative of the istance function. 5