2 NO Can you find the derivatives of the following functions? f(x) = x2 x4f(x) = 3x3 (2x2 – 3x + 1)f(x) = 5x8 exf(x) = x7 ln x** You can find the derivativefor problem 1 and 2 easily by multiplytwo functions, but not for problem3 and 4.** You can find the derivative for allof the above problems using the limitdefinition, but it can be a long andtedious processAs you know in section 10.5, the derivative of a sum is the sum of the derivatives.F(x) = u(x) + v(x)F’(x) = u’(x) + v’(x)Is the derivative of a product the product of the derivatives?NO
3 Derivatives of Products Theorem 1 (Product Rule)If f (x) = u(x) v(x), and if u ’(x) and v ’(x) exist, thenf ’ (x) = u(x) v ’(x) + u ’(x) v(x)In words: The derivative of the product of two functions is the first function times the derivative of the second function plus the second function times the derivative of the first function.
6 Example 3) Find f’(x) if f(x) = 5x8 ex f’(x) = 5x8 (ex)’ + ex (5x8)’f’(x) = 5x8 (ex) + ex (40x7)f’(x) = 5x8 ex x7 exf’(x) = 5x7 ex (x + 8) or 5x7 (x + 8) exNote that the only way to do is to apply the product rule
7 Example 4) Find f’(x) if f(x) = x7 ln x f’(x) = x7 (ln x)’ + ln x (x7)’f’(x) = x7 (1/x) + ln x (7x6)f’(x) = x x6 ln xf’(x) = x6 (1+ 7 lnx)
8 Example 5 Let f(x) = (2x+9)(x2 -12) A) Find the equation of the line tangent to the graph of f(x) at x = 3f’(x) = (2x+9)(2x) + (x2 – 12)(2)f’(x) = 4x2 + 18x + 2x2 – 24 = 6x2 + 18x - 24so slope m = f’(3) = 6(3)2 + 18(3) – 24 = 84also, if x = 3, f(3) = (2*3+9)(32 -12) = -45y = mx + b-45 = 84(3) + bb = - 297Therefore the equation of the line tangent is y = 84x - 297Find the values(s) of x where the tangent line is horizontalThe slope of a horizontal line is 0 so set f’(x) = 06x2 + 18x – 24 = 06(x2 + 3x – 4) = 06(x + 4) (x – 1) = 0 so x = -4 or 1
9 Derivatives of Quotients Theorem 2 (Quotient Rule)If f (x) = T (x) / B(x), and if T ’(x) and B ’(x) exist, thenIn words: The derivative of the quotient of two functions is the bottom function times the derivative of the top function minus the top function times the derivative of the bottom function, all over the bottom function squared.
15 Example 8The total sales S (in thousands of games) t months after the gameis introduced is given byA) Find S’(t)B) Find S(12) and S’(12). ExplainS(12) = and S’(12) = 2After 12 months, the total sales are 120,000 games. Sales areincreasing at a rate of 2000 games per month.C) Use the results above to estimate the total sales after 13 monthsS(13) = S(12) + S’(12) = 122. The total sales after 13 months is122,000 games.