1. 2.4 Derivatives of Trigonometric Functions

4. Example 1 Differentiate y = x 2 sin x. Solution: Using the Product Rule

5. Example 2 An object at the end of a vertical spring is stretched 4 cm beyond its rest position and released at time t = 0 (note that the downward direction is positive.) Its position at time t is s = f (t) = 4 cos t Find the velocity and acceleration at time t and use them to analyze the motion of the object.

6. Example 2 – Solution The velocity and acceleration are

7. Example 2 – Solution The object oscillates from the lowest point (s = 4 cm) to the highest point (s = –4 cm). The period of the oscillation is 2 , which is the period of cos t. cont’d

8. Example 3 – Solution The speed is | v | = 4 | sin t |, which is greatest when | sin t | = 1, that is, when cos t = 0. So the object moves fastest as it passes through its equilibrium position (s = 0). Its speed is 0 when sin t = 0, that is, at the high and low points. The acceleration a = –4 cos t = 0 when s = 0. It has greatest magnitude at the high and low points. cont’d

9. 2.5 The Chain Rule

11. Example Find F '(x) if F (x) =. Solution: (using the first definition) F (x) = (f  g)(x) = f (g(x)) where f (u) = and g (x) = x 2 + 1. Since and g(x) = 2x we have F (x) = f (g (x))  g (x)

12. Solution: (using the second definition) let u = x 2 + 1 and y =, then

13. The Chain Rule for powers:

14. Example Differentiate y = (x 3 – 1) 100. Solution: Taking u = g(x) = x 3 – 1 and n = 100 = (x 3 – 1) 100 = 100(x 3 – 1) 99 (x 3 – 1) = 100(x 3 – 1) 99  3x 2 = 300x 2 (x 3 – 1) 99

15. 2.6 Implicit Differentiation

16. So far we worked with functions where one variable is expressed in terms of another variable—for example: y = or y = x sin x (in general: y = f (x). ) Some functions, however, are defined implicitly by a relation between x and y, examples: x 2 + y 2 = 25 x 3 + y 3 = 6xy We say that f is a function defined implicitly - For example Equation 2 above means: x 3 + [f (x)] 3 = 6x f (x)

17. Example 1 For x 3 + y 3 = 6xy find:

18. Find the equation of the tangent line at (3,2) Example 2

19. Answer:

20. Practice problem: Find the slope of the curve at (4,4)

21. Answer: