2.1 Tangents and Derivatives at a Point

1. 2.1 Tangents and Derivatives at a Point

2. Finding a Tangent to the Graph of a Function To find a tangent to an arbitrary curve y=f(x) at a point P(x0,f(x0)), we • Calculate the slope of the secant through P and a nearby point Q(x0+h, f(x0+h)). • Then investigate the limit of the slope as h0.

3. Slope of the Curve If the previous limit exists, we have the following definitions. Reminder: the equation of the tangent line to the curve at P is Y=f(x0)+m(x-x0) (point-slope equation)

4. Example • Find the slope of the curve y=x2 at the point (2, 4)? • Then find an equation for the line tangent to the curve there. Solution (a) (b) The equation is y=4+4(x-2), that is, y=4x-4.

5. Derivative of a Function f at a Point x0 The expression is called the difference quotient of f at x0 with increment h. If the difference quotient has a limit as h approaches zero, that limit is named below.

6. 2.2 The Derivative as a Function We now investigate the derivative as a function derived from f by Considering the limit at each point x in the domain of f. If f’ exists at a particular x, we say that f is differentiable (has a derivative) at x. If f’ exists at every point in the domain of f, we call f is differentiable.

7. Alternative Formula for the Derivative An equivalent definition of the derivative is as follows. (let z = x+h)

8. Calculating Derivatives from the Definition The process of calculating a derivative is called differentiation. It can be denoted by Example. Differentiate Example. Differentiate for x>0.

9. Notations There are many ways to denote the derivative of a function y = f(x). Some common alternative notations for the derivative are To indicate the value of a derivative at a specified number x=a, we use the notation

10. Differentiable on an Interval; One-Sided Derivatives If a function f is differentiable on an open interval (finite or infinite) if it has a derivative at each point of the interval. It is differentiable on a closed interval [a, b] if it is differentiable on the interior (a, b) and if the limits exist at the endpoints. Right-hand derivative at a Left-hand derivative at b A function has a derivative at a point if and only if the left-hand and right-hand derivatives there, and these one-sided derivatives are equal.

12. When Does A Function Not Have a Derivative at a Point A function can fail to have a derivative at a point for several reasons, such as at points where the graph has • a corner, where the one-sided derivatives differ. • a cusp, where the slope of PQ approaches  from one side and -  from the other. • a vertical tangent, where the slope of PQ approaches  from both sides or approaches -  from both sides. • a discontinuity.

14. Differentiable Functions Are Continuous Note: The converse of Theorem 1 is false. A function need not have a derivative at a point where it is continuous. For example, y=|x| is continuous at everywhere but is not differentiable at x=0.

15. 2.3 Differentiation Rules

16. The Power Rule is actually valid for all real numbers n.

17. Example.

18. Note: Example.

19. Example.

21. Example Example: Solution:

23. Example Example: Solution:

24. Higher derivatives The derivative f’ of a function f is itself a function and hence may have a derivative of its own. If f’ is differentiable, then its derivative is denoted by f’’. So f’’=(f’)’ and is called the second derivative of f. Similarly, we have third, fourth, fifth, and even higher derivatives of f.

25. A general nth order derivative can be denoted by Example:

26. 2.5 Derivatives of Trigonometric Functions

27. Example Example: Solution:

28. Example Example: Solution:

30. Example Example: Solution:

31. 2.6 Exponential Functions In general, if a1 is a positive constant, the function f(x)=ax is the exponential function with base a.

32. If x=n is a positive integer, then an=a  a  …  a. If x=0, then a0=1, If x=-n for some positive integer n, then If x=1/n for some positive integer n, then If x=p/q is any rational number, then If x is an irrational number, then

33. Rules for Exponents

34. The Natural Exponential Function ex The most important exponential function used for modeling natural, physical, and economic phenomena is the natural exponential function, whose base is a special number e. The number e is irrational, and its value is 2.718281828 to nine decimal places.

35. The graph of y=ex has slope 1 when it crosses the y-axis.

36. Derivative of the Natural Exponential Function Example. Find the derivative of y=e-x. Solution:

37. 2.7 The Chain Rule

38. Example Example: Solution:

39. “Outside-inside” Rule It sometimes helps to think about the Chain Rule using functional notation. If y=f(g(x)), then In words, differentiate the “outside” function f and evaluate it at the “inside” function g(x) left alone; then multiply by the derivative of the “inside” function.

40. Example Example. Differentiate sin(2x+ex) with respect to x. Solution. Example. Differentiate e3x with respect to x. Solution.

41. In general, we have For example.

42. Repeated Use of the Chain Rule Sometimes, we have to apply the chain rule more than once to calculate a derivative. Example. Solution.

43. The Chain Rule with Powers of a Function If f is a differentiable function of u and if u is a differentiable function of x, then substituting y = f(u) into the Chain Rule formula leads to the formula This result is called the generalized derivative formula for f. For example. If f(u)=un and if u is a differentiable function of x, then we can Obtain the Power Chain Rule:

44. Example Example: Solution:

45. Example Example: Solution:

46. Example Example: Solution:

47. 2.8 Implicit Differentiation