In this chapter, we begin our study of differential calculus.

1. In this chapter, we begin our study of differential calculus. • This is concerned with how one quantity changes in relation to another quantity.

2. DERIVATIVES the derivative is a measure of how fast does a function change in response to changes in independent variable; for example, the derivative of the position of a moving object with respect to time is the object's velocity.

3. Definition of INFINITESIMAL 1:taking on values arbitrarily close to but greater than zero 2:immeasurably or incalculably small <an infinitesimal difference> Now concentrate and observe:

4. Let’s assume that we know function at any →graph. Let’s find the slope of the line joining position P(x = a) and some T . = T f (x+Δx) - f(x) P x

5. f (x) x Let us consider Δ x intervals that are getting smaller and smaller. Df = f (x+Δx) - f(x) Dx S Q T R f (x+Δx) - f(x) P D x ALTHOUGH, both, Δf andΔxare becoming infinitesimally small approaching zero, their ratio is approaching definite value – slope of tangent line at P Slopes of straight lines connecting point P and other points on the path are approaching the slope of the line tangent at P.

6. Explain yourself and me what you saw

7. “The derivative of fwith respect to xis …” Definition: The derivative of a function at a fixed number is provided that this limit exists. Very often you are going to find that definition in following form: Graphical interpretation of mathematical definition of derivative at point is the slope of the tangent line to the function at point .

8. Till now, we considered the derivative of a function at a fixed number . Now, we change our point of view and let the number vary. Let’s assume that can take any value of on an open interval Definition: Let be a function. The derivative of is the function whose value at x is the limit provided this limit exists. If this limit exists for each x in an open interval I, then we say that f is differentiable on I. The derivative is the instantaneous rate of change of a function with respect its variable. This is equivalent to finding the slope of the tangent line to the function at a point.

9. There are many ways to write the derivative of “f prime x” or “the derivative of f with respect to x” “y prime” “the derivative of y with respect to x” or “d y d x” “d f d x” “the derivative of f with respect to x” or “the derivative of f of x with respect to x” or “d dx of f of x”

10. Rates of change • This means that: • When the derivative is large (and therefore the curve is steep, as at the point P in the figure), the y-values change rapidly. • When the derivative is small, the curve is relatively flat and the y-values change slowly.

11. The derivative is defined at the end points of a function on a closed interval. The derivative is the slope of the original function.

13. Differentiability A function is differentiable if it has a derivative everywhere in its domain. To be differentiable, a function must be continuous and smooth. Functions on closed intervals must have one-sided derivatives defined at the end points. Derivatives will fail to exist at: A function has derivative at a point, if the left derivative is equal to the right derivative at that point. The left slope must be equal to the right slope. cusp corner discontinuity vertical tangent

14. Find an equation of the tangent line to the hyperbola at the point (3, 1). • The slope of the tangent at (3, 1) is: Eq. of the tangent at the point (3, 1) is x + 3y – 6 = 0 The hyperbola and its tangent are shown in the figure How do you find normal at (3,1) ?

15. Equation of the tangent line to a function at the point Equation of the normal line to a function at the point

16. We can estimate the value of the derivative at any value of x by drawing the tangent at the point (x,f(x)) and estimating its slope. For instance, for x = 5, we draw the tangent at P in the figure and estimate its slope to be about 3/2, so . This allows us to plot the point P’(5, 1.5) on the graph of f’ directly beneath P. Repeating this procedure at several points, we get the graph shown in this figure. Tangents at x = A, B, and C are horizontal. • So, the derivative is 0 there and the graph • of f’ crosses the x-axis at those points. Between A and B, the tangents have positive slope. So, f’(x) is positive there. Between B and C, and the tangents have negative slope. So, f’(x) is negative there.

17. example: Find Sketch both graphs. • Notice that = 0 when f has horizontal tangents and is positive when the tangents have positive slope.

18. THE DERIVATIVE AS A FUNCTION Example 2 b • We use a graphing device to graph f • and f’ in the figure. • Notice that f’(x) = 0 when f has horizontal tangents and f’(x) is positive when the tangents have positive slope. • So, these graphs serve as a check on our work in part (a).

19. If fis differentiable at x = c, then f is continuous at x = c. Two theorems: 1. Differentiability Implies Continuity Since a function must be continuous to have a derivative, if it has a derivative then it is continuous. The converse: "If a function is continuous at c, then it is differentiable at c," - is not true. This happens in cases where the function "curves sharply." Differentiability implies continuity, continuity doesn’t imply differentiability.

20. Intermediate Value Theorem for Continuous Functions If is continuous on and is any number between and , then there is at least one number such that . example: Prove that function has a root/zero between 2 and 2.5. is continuous on ], and , so must have a zero between 2 and 2.5.

21. Intermediate Value Theorem for Derivatives Let be differentiable on and suppose that k is a number between and . Then there exists a point such that . You can find it in this form too: If and are any two points in an interval on which is differentiable, then takes on every value between and . Between and , must take on every value between ½ and 3. is continuous function on

22. How to find derivatives???

23. If it were always necessary to compute derivatives directly from the definition, calculus BC would be even worse nightmare, there would be no Ipad and the life as we know it ……. Computations would be tedious, and the evaluation of some limits would require ingenuity. • Fortunately, several rules have been developed for finding derivatives without having to use the definition directly. • These formulas greatly simplify the task of differentiation. First let us see what are we avoiding: Let’s start with the simplest of all functions — the constant function f(x) = c.

24. The graph of this function is the horizontal line y = c, which has slope 0. In Leibniz notation The derivative of a constant is zero examples:

25. (Pascal’s Triangle) We observe a pattern: …

26. We observe a pattern: … examples: power rule

27. examples: limit has no effect on a constant coefficient; the constant could be factored to the outside.

28. limit of the sum = sum of the limits provided they are both differentiable each term is treated separately examples:

29. example:

30. Horizontal tangents occur when slope = zero. Example: Find the horizontal tangents of: (The function is even, so we only get two horizontal tangents.)

31. The derivative of a product of two functions is • the first function times the derivative of the second function plus • the second function times the derivative of the first function, • provided they are both differentiable.

32. Example: Find derivative if • Later, though, we will meet functions, such as y = x2 sinx, for which the product rule is the only possible method.

33. Q

34. Q • The derivative of a quotient is the denominator times the derivative of the numerator minus the numerator times the derivative of the denominator, all divided by the square of the denominator, • provided they are both differentiable.

35. The theorems of this section show that: • Any polynomial is differentiable on . • Any rational function is differentiable on its domain. Furthermore, the Quotient Rule and the other differentiation formulas enable us to compute the derivative of any rational function—as the next example illustrates. Example:

36. Don’t use the Quotient Rule every time you see a quotient. • Sometimes, it’s easier to rewrite a quotient first to put it in a form that is simpler for the purpose of differentiation. • For instance: • It is possible to differentiate the function • using the Quotient Rule. • However, it is much easier to perform the division first and write the function as before differentiating.

37. The Quotient Rule can be used to extend the Power Rule to the case where the exponent is a negative integer. If n is a positive integer, then Example:

38. So far, we know that the Power Rule holds if the exponent n is a positive or negative integer. • If n = 0, then x0 = 1, which we know has a derivative of 0. • Thus, the Power Rule holds for any integer n. What if the exponent is a fraction? In fact, it can be shown by using Chain Rule (obviously proof later) that it also holds for any real number n. If n is any real number, then

39. Example:

40. Example: Find equations of the tangent line and normal line to the curve In your mind: at the point (1, ½). slope of the tangent line at (1, ½) : tangent line at (1, ½): normal line at (1, ½):

41. Example: At what points on the hyperbola xy = 12 is the tangent line parallel to the line 3x + y = 0? • Since xy = 12 can be written as y = 12/x, we have: • Let the x-coordinate of one of the points in question be . • Slope of the tangent line at that point is , and that has to be equal to the slope of line 3x + y = 0 • the required points are: (2, 6) and (-2, -6)

42. Here’s a summary of the differentiation formulas we have learned so far.

43. is the first derivative of y with respect to x. is the second derivative. is the third derivative. is the fourth derivative. Higher Order Derivatives: (y double prime) We will learn later what these higher order derivatives are used for. p