Chapter 3 Additional Derivative Topics

1. Chapter 3Additional DerivativeTopics Section 3 Derivatives of Products and Quotients

2. Derivatives of Products In Section 2.5, we found that the derivative of a sum is the sum of the derivatives and the derivative of a difference is the difference of the derivatives. In this section, we consider if similar properties hold for multiplication and division of functions. We first “test” the theory with an example.

3. Derivatives of Products Let f(x) = 3x + 2 and g(x) = 2x – 3. Determine if the derivative of the product (f(x)·g(x))´is equal to the product of the derivatives, f ´(x)·g ´(x). Solution: Find the product, f(x)·g(x). f(x)·g(x) = (3x + 2)(2x – 3) = 6x2 – 5x – 6 Find the derivative of the product (f(x)·g(x))´. (f(x)·g(x))´ = 12x – 5 Find the derivatives f ´(x) and g ´(x). f ´(x) = 3 and g ´(x) = 2 f ´(x)·g ´(x) = 3·2 = 6 and (f(x)·g(x))´ = 12x – 5 Therefore, (f(x)·g(x))´ ≠ f ´(x)·g ´(x). The derivative of a product is not the product of the derivatives.

4. Derivatives of Products We found that the derivative of the product of two functions is not the product of the derivatives of the two functions. Using the definition of the derivative and the four-step process, we can show that • The derivative of the product of two functions is the first times the derivative of the second, plus the second times the derivative of the first.

5. Theorem 1 Product Rule • If y = f(x) = F(x)·S(x) • (where F represents the First function in a product and S represents the Second function in a product), • and if F ´(x) and S´(x) exist, then • f ´(x) = F(x)·S´(x) + S(x)·F´(x). • Using simplified notation, • y´ = FS´ + SF´ or

6. Example Differentiating a Product Use two different methods to find f ´(x) for f(x) = 2x2(3x4 – 2). Solution: Method 1 Use the product rule with F(x) = 2x2 and S(x) = 3x4 – 2. f ´(x) = 2x2(3x4 – 2)´ + (3x4 – 2)(2x2)´ = 2x2(12x3) + (3x4 – 2)(4x) = 24x5 + 12x5 – 8x f ´(x) = 36x5 – 8x Method 2 Multiply first, then find the derivative. f(x) = 2x2(3x4 – 2) = 6x6 – 4x2 f ´(x) = 36x5 – 8x

7. Example Tangent Lines Let f(x) = (2x – 9)(x2 + 6). (A) Find the equation of the line tangent to the graph of f(x) at x = 3. Solution: Find the derivative f ´(x). f ´(x) = (2x – 9)(x2 + 6)´ + (x2 + 6)(2x – 9)´ = (2x – 9)(2x) + (x2 + 6)(2) Note: In this example, since we seek the slope of the tangent line at x = 3 given by f ´(3), it is not required that we simplify the derivative expression. f ´(3) = (2·3 – 9)(2·3) + (32 + 6)(2) = –18 + 30 = 12 The tangent to f(x) at x = 3, has slope 12.

8. Example Tangent Lines Continued Let f(x) = (2x – 9)(x2 + 6). (A) Find the equation of the line tangent to the graph of f(x) at x = 3. Solution: We found that the tangent to f(x) at x = 3, has slope 12. Evaluate f(3) = (2·3 – 9)(32 + 6) = (–3)(15) = –45. Use the slope m = 12 and the point (x1, y1) = (3, –45) in the point-slope form for a line y – y1 = m(x – x1). y – (–45) = 12(x – 3) y = 12x – 81

9. Example Tangent Lines Let f(x) = (2x – 9)(x2 + 6). (B) Find the value(s) of x where the tangent line is horizontal. Solution: The tangent line is horizontal when its slope is zero. We found the derivative to be f ´(x) = (2x – 9)(2x) + (x2 + 6)(2). We solve f ´(x) = (2x – 9)(2x) + (x2 + 6)(2) = 0. 6x2 – 18x + 12 = 0 x2 – 3x + 2 = 0 (x – 1)(x – 2) = 0 x = 1, 2 The tangent line is horizontal when x = 1 and x = 2.

10. The Derivative of the Quotient of Two Functions Consider the functions T(x) = x5 and B(x) = x2. T´(x) = 5x4 and B´(x) = 2x. The derivative of a quotient is not equal to the quotient of the derivatives.

11. Derivative of the Quotient of Two Functions The previous example shows that the derivative of the quotient of two functions is not the quotient of their derivatives. The derivative of a quotient is a more complicated quotient than expected. If T(x) and B(x) are any two differentiable functions (where T is the Top function and B is the Bottom function) The derivative of the quotient of two functions is the denominator times the derivative of the numerator, minus the numerator times the derivative of the denominator, divided by the denominator squared.

12. Theorem 2 Quotient Rule If T(x) and B(x) are any two differentiable functions (where T is the Top function and B is the Bottom function)

13. Example Differentiating Quotients

14. Example Differentiating Quotients

15. Example Sales Analysis The total sales S (in thousands of games) of a video game t months after the game is introduced are given by (A) Find S´(t).

16. Example Sales Analysis The total sales S (in thousands of games) of a video game t months after the game is introduced are given by (B) Find S(10) and S´(10). Interpret these results. Total sales after 10 months are 62,500 games. Sales are increasing at a rate of 6,250 games per month.

17. Example Sales Analysis The total sales S (in thousands of games) of a video game t months after the game is introduced are given by (C) Use the results from part (B) of this example to estimate the total sales after 11 months. Since S(10) = 62,500 and S´(10) = 6,250, the estimated sales after 11 months are 62,500 + 6,250 = 68,750 games.