1. Chapter 2 Derivatives up down return end

2. 2.6 Implicit differentiation 1) Explicit function: The function which can be described by expressing one variable explicitly in terms of another variable (other variables) are generally called explicit function---for example, y=xtanx, or y=[1+x 2 +x 3 ] 1/2, or in general y=f(x). 2) Implicit function: The functions which are defined implicitly by a relation between variables--x and y--are generally called implicit functions--- such as x 2 +y 2 =4, or 7sin(xy)=x 2 +y 3 or, in general F(x,y)=0.If y=f(x) satisfies F(x, f(x))=0 on an interval I, we say f(x) is a function defined on I implicitly by F(x,y)=0, or implicit function defined by F(x,y)=0. up down return end

3. 3) Derivatives of implicit function Suppose y=f(x) is an implicit function defined by sin(xy)= x 2 +y 3. T hen sin[xf(x)] =x 2 + [f(x)] 3. From the equation, we can find the derivative of f(x) even though we have not gotten the expression of f(x). Fortunately it is not necessary to solve the equation for y in terms of x to find the derivative. We will use the method called implicit differentiation to find the derivative. Differentiating both sides of the equation, we obtain that [f(x)+xf ' (x)]cos [xf(x)] =2x+3 [f(x)] 2 f ' (x). Then up down return end

4. Example (a) If x 3 +y 3 =27, find (b) Find the equation of the tangent to the curve x 3 +y 3 =28 at point (1,3). up down return end

5. Example (a) If x 3 +y 3 =6xy, find y'. (b) Find the equation of the tangent to the folium of Descartes x 3 +y 3 =6xy at point (3,3). up down return end

6. Orthogonal: Two curves are called Orthogonal, if at each point of intersection their tangent lines are perpendicular. If two families of curves satisfy that every curve in one family is orthogonal to every curve in the another family, then we say the two families of curves are orthogonal trajectores of each other. Example The equations xy=c (c  0) represents a family of hyperbolas. And the The equations x 2 -y 2 =k (k  0) represents another family of hyperbolas with asymptotes y=  x. Then the two families of curves are trajectores of each other. up down return end

7. Derivative f' (x) of differentiable function f(x) is also a function. If f' (x) is differentiable, then we have [f ' (x)] '. We will denote it by f ' ' (x), i.e., f' ' (x)=[f ' (x)] '. The new function f ' ' (x) is called the second derivative of f(x). If y=f(x), we also can use other notations: Si milarly f ' ' ' (x)=[f '' (x)] ' is called the third derivative of f(x), and 2.7 Higher derivatives up down return end

8. And we can define f' ' ' ' (x)=[f ' ' ' (x)] '. From now on instead of using f' ' ' ' (x) we use f (4) (x) to represent f ' ' ' ' (x). In general, we define f (n) (x)=[f (n-1) (x)] ', which is called the nth derivative of f(x). We also like to use the following notations, if y=f(x), Example If y=x 4 -3x 2 +6x+9, find y ', y ' ', y ' ' ', y (4). up down return end Example If f(x)=, find f (n) (x). Example If f(x)=sinx, g(x)=cosx, find f (n) (x) and g (n) (x). Example Find y ' ', if x 4 +y 3 =x-y.

9. 2.8 Related rates (omitted)

10. 2.9 Differentials, Linear and Quadratic Approximations Definition : Let  x=x-x 0,  f(x) =f(x)-f(x 0 ). If there exists a constant A(x 0 ) which is independent of x and  x such that  f(x)=A(x 0 )  x+B(x, x 0 ) where B(x, x 0 ) satisfies. Then A  x is called differential of f(x) at x 0. Generally A  x is denoted by df(x)| x=x 0 = A(x 0 )  x. Replacing x 0 by x, the differential is denoted by df(x) and df(x)= A(x)  x. up down return end

11. Proof: From the definition, Corollary: If the differential of f(x) is df(x)= A(x)  x, then f(x) is differentiable and A(x)=f '(x). Corollary: (a) If f(x)=x, then dx=df(x)=  x. (b) If f(x) is differentiable, then differential of f(x) exists and df(x)=f '(x)dx. up down return end

12. Example (a) Find dy, if y=x 3 +5x 4. (b) Find the value of dy when x=2 and dx=0.1. Solution: Geometric meaning of differential of f(x), df(x)=QS  f(x)=RS x ox y P t S R Q dx=  x dy y=f(x) As  x=dx is very small,  y=dy,i.e., f(t)-f(x) f ' (x)  t. up down return end

13. Example Use differentials to find an approximate (65) 1/3. From definition of the differential, we can easily get If f(x) is differentiable at x=a, and x is very closed to a, then f(x) f(a)+f ' (a)(x-a). The approximation is called Linear approximation or tangent line approximation of f(x) at a. And function L(x)= f(a)+f ' (a)(x-a) is called the linearization of f(x) at a. up down return end

14. Example Find the linearization of the function f(x)=(x+3) 1/2 and approximations the numbers (3.98) 1/2 and (4.05) 1/2. up down return end

15. Quadratic approximation to f(x) near x=a : S uppose f(x) is a function which the second derivative f ' '(a) exists. P(x)=A+Bx+Cx 2 is the parabola which satisfies P(a)=f(a), P '(a)=f '(a), and P ' '(a)=f ' '(a). As x is very closed to a, the P(x) is called Quadratic approximation to f(x) near a. Corolary: S uppose P(x)=A+Bx+Cx 2 is the Quadratic approximation to f(x) near a. Then P(x)=f(a)+f '(a)(x-a)+ f' '(a)(x-a) 2 / 2. If P(x) is the quadratic approximation to f(x) near x=a, then as x is very closed to a, P(x) f(x).That is f(x) f(a)+f ' (a)(x-a)+ f ' ' (a)(x-a) 1/2 /2. up down return end

16. Example Find the quadratic approximation to f(x)=cosx near 0.0. up down return end

17. Example Find the quadratic approximation to f(x)=(x+3) 1/2 near x=1. up down return end

18. The method is to give a way to get a approximation to a root of an equation. 2.10 Newton’s method(to be omitted) Suppose f(x) is defined on [a,b], f ' (x) does not value 0. Let x 0  [a,b], f(a)f(b)<0. And x 1 =x 0 -, x 2 =x 1 -. Keeping repeating the process (x n =x n-1 - ), we obtain a sequence of approximations x 1, x 2,..., x n,...... If, then r is the root of the equation f(x)=0. up down return end

19. Example Starting with x 1 =2, find the third approximation x 3 to the root of the equation x 3 -2x-5=0.

20. 2.1 Derivatives We defined the slope of the tangent to a curve with equation y=f(x) at the point x=a to be Generally we give the following definition: up down return end

21. Definition: The derivative of a function f at a number a, denoted by f´(a), is if this limit exists. Then we have: up down return end

22. Example Find the derivative of the function y=x 2 -8x+9 at a. Geometric interpretation: The derivative of the function y=f(x) at a is the slope of tangent line to y=f(x) at (a, f(a)). The line is through (a, f(a)).So if f ´(a) exists, the equation of the tangent line to the curve y=f(x) at (a, f(a)) is y-f(a)= f ´(a) (x-a). up down return end

23. Example Find the equation of the tangent line of the function y=x 2 -3x+5 at x=1. In the definition if we replace a by x, then we obtain a new function f ´(x) which is deduced from f(x). up down return end

24. Example If f(x)=(x-1) 1/2, find the derivative of f. State the domain of f´(x). Example F ind the derivative of f if 1-x f(x)= 2+x Other notations: If y=f(x), then the other notations are that f´(x)= y´= = = =Df(x)=D x f(x). up down return end

25. The symbol D and d/dx are called differential operators. We also use the notations: Definition A function f is called differentiable at a if f´(a) exists. It is differentiable on an open interval (a,b) [or (a,+  ) or (- ,b) ] if it is differentiable at every number in the interval. Example Where is the function f(x)=|x| is differentiable? up down return end

26. Theorem: If f(x) is differentiable at a, then f(x) is continuous at a.( The converse is false) (3) the points at which the curve has a vertical tangent line, such as, f(x)=x 1/3, at x=0. (1) the points at which graph of the function f has “corners”, such as f(x)=|x| at x=0; (2) the points at which the function is not continuous, such as, the function, defined as f(x)=2x for x  1, and 3x for x<1, at x=1; There are several cases a function fails to be differentiable up down return end

27. 2.2 Differentiation 1). Theorem If f is a constant function, f(x)=c, then f´(x)= (c)´=0, i.e., =0. up down return end

28. 2). The power rule If f(x)=x n, where n is a positive integer, then f´(x)= nx n-1, x n =nx n-1. Example If f(x)=x 100, find f´(x). up down return end

29. 3)Theorem Suppose c is a constant and f´(x) and g´(x) exist.Then Example If f(x)= x 50 +x 100, find f´(x). (c) (f(x)-g(x))´exists and (f(x)-g(x)´=f´(x)-g´(x). (b) (f(x)+g(x))´exists and (f(x)+g(x)´=f´(x)+g´(x); (a) (cf(x))´ exists and (cf(x))´=cf´(x); up down return end

30. 4) Product rule Suppose f´(x) and g´(x) exist. Then f(x)g(x) is differentiable and [f(x)g(x)]´= f´(x) g(x)+f(x)g´(x). Example If f(x)= (2x 5 )(3x 10 ), find f´(x). up down return end

31. 4) Quotient rule Suppose f´(x) and g´(x) exist and g(x)  0, then f(x)/g(x) is differentiable and [f(x)/g(x)]´= [f´(x) g(x)-f(x)g´(x)]/[g(x)] 2. Example If f(x)=, find f´(x). x 2 +2x-5 x 3 -6 up down return end

32. 2). The power rule (general version) If f(x)=x n, where n is any real number, then f´(x)= nx n-1,,i.e., x n =nx n-1. Example If f(x)=x , find f´(x). If g(x)= x 1/2, g´(x)=? up down return end Example Differentiate the function f(t)=(1-t)t 1/3. Table of differentiation formulas (in paper 119)

33. 2.3 Rate of change in the Economics Suppose C(x) is the total cost that a company CC xx = C(x 2 )-C(x 1 ) x 2 -x 1 = C(x 1 +  x)-C(x 1 )  x average of change of the cost is the additional cost is  C= C(x 2 )-C(x 1 ), and the number of items produced increased from x 1 to x 2, The function C is called a cost function. If the incurs in producing x units of certain commodity. up down return end

34. The limit of this quantity as  x  0, is called the marginal cost by economist. Marginal cost= Taking  x=1 and n large (so that  x is small compared to n),we have C'(n) C(n+1)-C(n). Thus the marginal cost of producing n is approximately equal to the cost of producing one more unit [the (n+1)st unit]. up down return end

35. 2.4 Derivatives of trigonometric functions (1) Theorem Proof: suppose OP=1 and  (0,  /2). So we will show A y x C B o  D Notice that 0<|BC|<arcAB up down return end (2) Corollary

36. (3) T heorem x C B o  D y A Proof: Notice that Area of  OAB <Area of sector OAB<Area of  OAD. up down return end (4) Corollary

37. E xample Find up down return end

38. (5) T heorem E xample Differentiate y=xsinx. up down return end (6) T heorem E xample Differentiate y=tanx. Corollary (tanx)'=sec 2 x

39. E xample Differentiate y=cotx. Corollary (cotx)'= - csc 2 x E xample Differentiate f(x)= up down return end

40. 2.5 Chain rule The chain rule If the derivative g'(x) and derivative f '(u), with respect to u, exist, then the composite function f(g(x)) is differentiable, and [f(g(x))] ' =f '(g(x))g '(x). Let  u=g(x+  x)-g(x)  y=f(u+  u)-f(u) Proof: up down return end

41. Case 1: du/dx  0, then  u  0 Case 2: du/dx=0. there are two cases: (a) u u 0, (b)  u= 0, up down return end

42. E xample Find F '(x) if F(x)=(1+x 2 ) 3/4. up down return end