Calculus on the wall

1. mastermathmentor.com presents Calculus on the wall Helping students learn … and teachers teach 3. Differentiation Created by: Stu Schwartz Un-narrated Version Edited by: Jake Weinberg Graphics: Apple Grapher: Version 2.3 Math Type: Version 6.7 Intaglio: 2.9.5a Fathom: Version 2.11

2. Derivative Rules www.mastermathmentor.com

3. Vocabulary Terms used in this slide show: • Derivative • Differentiation • Constant rule • Power rule • Constant multiple rule • Sum and difference rules • Splitting • Tangent line • Normal line • Product rule • Quotient rule • Chain rule • Higher-order derivatives • Explicit differentiation • Implicit differentiation www.mastermathmentor.com

4. Finding the Slope of the Tangent Line Given the function y = f(x) and points P(x0, y0),Q(x1, y1) on f, we define the slope of the secant line: y1— y0 h = x1 — x0 Since h = x1 — x0, x1 = x0 + h We define the slope of the tangent line to f at x0: www.mastermathmentor.com

5. Three Important Formulas The slope of the secant line on f between (x0, y0) and (x1, y1): The slope of the tangent line to f at (x0, y0): The equation of the tangent line to f at (x0, y0) — The point-slope formula from algebra: www.mastermathmentor.com

6. Derivative Definition To find the slope of the tangent to f(x) at any value x, we define the derivative of the function f’(x): The derivative is simply a formula for the slope of the tangent line to the function f at any x-value. The process of taking derivatives is called differentiation.There are several notations for derivatives: www.mastermathmentor.com

7. Derivative Rules There are a series of rules that make taking derivatives much simpler than using the definition of the derivative. The Constant Rule www.mastermathmentor.com

8. The Power Rule The Power Rule Iff(x) = xn , then f’(x) = n•xn-1. When n = 1, we are taking the derivative of f(x) = x1. By the power rule, f’(x) = 1x0 = 1. This is consistent with the fact that the slope of the line y = x is 1. www.mastermathmentor.com

9. Using the Power Rule Using the power rule is straightforward for expressions like y = x3, but many expressions need to be rewritten in order to use the power rule and may need to be simplified after they are differentiated. www.mastermathmentor.com

10. Multiplying by a Constant Constant Multiple Rule: 1 2 3 4 5 6 www.mastermathmentor.com

11. Sum and Difference Formulas The derivative of a sum or difference is the sum or difference of its derivatives. www.mastermathmentor.com

12. Using the Derivative Find the equation of the tangent line and normal line (perpendicular) to the given curve at the given x-value. www.mastermathmentor.com

13. Linear Approximation A linear approximation is an approximation of a differentiable function f using a linear function at a point close to x = a. We want the value of f(c) – the blue dot. When c is close to a, the red dot will be a good approximation to f(c). www.mastermathmentor.com

14. The Chain Rule Given y = f(x), find f’(x) and f’(2) www.mastermathmentor.com

15. The Product Rule The Product Rule www.mastermathmentor.com

16. The Quotient Rule The Quotient Rule www.mastermathmentor.com

17. Derivative of Trigonometric Functions You must know how to take the derivative of trig functions. It is best to memorize this chart. If u is a function of x, these trig derivative forms are in this table. The Trig Rules www.mastermathmentor.com

18. Forms of Expressions www.mastermathmentor.com

19. Problems Determine the form of the problem and then take the derivative. www.mastermathmentor.com

20. More Problems www.mastermathmentor.com

21. Simplification Your teacher will specify the amount of simplification necessary, but when answering AP multiple choice questions, you have to match answers. Additional Simplification Minimum work www.mastermathmentor.com

22. More Simplifications Minimum work Additional Simplification www.mastermathmentor.com

23. Using a Table Tables are sometimes given so that calculator technology cannot be used to find derivatives at given values of x. Use the table to find h’(2) www.mastermathmentor.com

24. More Table Problems Use the tableto find h’(2) www.mastermathmentor.com

25. Higher-Order Derivatives A second derivative of a function is the derivative of its derivative. The third derivative is the derivative of the second derivative and so on. There are various notations for higher-order derivatives. www.mastermathmentor.com

26. Implicit Differentiation When a function is written as a function of x, it is said to be defined explicitly. When an expression is not in the form: y = , but is defined in terms of an algebraic relationship, it is said to be defined implicitly. Expression defined implicitly Expression defined explicitly When you differentiate terms involving x alone, you differentiate as usual, but when you differentiate terms involving y, you must apply the chain rule as y is defined in terms of x. www.mastermathmentor.com

27. Implicit Differentiation • Differentiate both sides of the equation with respect to x. • If you are finding dy/dx at a point, plug in the point and solve for dy/dx. • If you are finding dy/dx, collect all terms involving dy/dxon one side of the equation and move all other terms to the other side. • Factor dy/dx and divide both sides by the other factor. www.mastermathmentor.com

28. Implicit Differentiation www.mastermathmentor.com

29. Summary www.mastermathmentor.com

30. Vocabulary Do you understand each term? • Derivative • Differentiation • Constant rule • Power rule • Constant multiple rule • Sum and difference rules • Splitting • Tangent line • Normal line • Product rule • Quotient rule • Chain rule • Higher-order derivatives • Explicit differentiation • Implicit differentiation www.mastermathmentor.com