1. Wednesday, February 17, 2016MAT 145

2. Wednesday, February 17, 2016MAT 145 THE CHAIN RULE WORDS BY: JOHN A. CARTER TUNE: "CLEMENTINE" Here's a function in a function And your job here is to find The derivative of the whole thing With respect to x inside. Call the outside f of u And call the inside u of x. Differentiate to find df/du And multiply by du/dx. Use the chain rule. Use the chain rule when e'er you find The derivative of a function compositionally defined.

3. Wednesday, February 17, 2016MAT 145

4. Wednesday, February 17, 2016MAT 145

5. Wednesday, February 17, 2016MAT 145 An implicitly defined function is: A function whose relation among the variable is given by an equation for which the function has not been explicitly stated. In the equation x 2 + y 2 = 25, y is an implicit function of x because the equation doesn’t explicitly express y in terms of x.

6. Wednesday, February 17, 2016MAT 145

7. Wednesday, February 17, 2016MAT 145 Take derivative of each term. Use chain rule. X changes identically to itself; dx/dx=1. Solve for dy/dx. Simplify.

8. Wednesday, February 17, 2016MAT 145 To calculate the derivative of an implicitly defined function: Assume a functional connection among the variables. If x and y are present, assume y is a function of x. Calculate the derivative of each term in the equation. Because we don’t explicitly know how y is determined by x, when calculating the derivative of y, we use the chain rule and write dy / dx as the derivative of y. After determining term-by-term derivatives, carry out all necessary algebra steps to isolate dy / dx. That’s our goal!

9. Wednesday, February 17, 2016MAT 145

10. Wednesday, February 17, 2016MAT 145

11. Wednesday, February 17, 2016MAT 145 A rabbit moves within a straight horizontal tube according to the position function such that s(t) is measured in feet and t in seconds. When s(t) is positive, the rabbit is to the right of some arbitrary point labeled 0. 1.Calculate the rabbit’s velocity and acceleration functions over the interval from 0 to 6 seconds. 2.Determine the times from 0 to 6 seconds when the rabbit is moving to the right and when it is moving to the left. Explain your response with justification. 3.Determine when the rabbit is speeding up and when it is slowing down. Explain your response with justification.

12. Wednesday, February 17, 2016MAT 145

13. Wednesday, February 17, 2016MAT 145

14. Wednesday, February 17, 2016 MAT 145

15. Wednesday, February 17, 2016MAT 145

16. Wednesday, February 17, 2016MAT 145

17. Wednesday, February 17, 2016MAT 145 A particle moves on a number line with its position given by such that s(t) is measured in cm and t in seconds. 1. Determine the particle’s velocity and acceleration functions and then sketch graphs of position, velocity, and acceleration on the interval from 0 to 6 seconds. 2. Determine the times from 0 to 6 seconds when the particle is moving to the right, when it is moving to the left, and when it is at rest. Use interval notation or inequality notation. Explain and justify your response. 3. Determine when the particle is speeding up and when it is slowing down. Use interval notation or inequality notation. Explain and justify your response.