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Chapter 2 Derivatives

Chapter 2 Derivatives. up down return end. 2.6 Implicit differentiation.

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Chapter 2 Derivatives

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  1. Chapter 2 Derivatives up down return end

  2. 2.6Implicit 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+x2+x3]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 x2+y2 =4, or 7sin(xy)=x2+y3or, 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)= x2+y3. Then sin[xf(x)]=x2+ [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)]2f '(x). Then up down return end

  4. Example (a) If x3+y3 =27, find (b)Find the equation of the tangent to the curve x3+y3 =28 at point (1,3). up down return end

  5. Example (a) If x3+y3 =6xy, find y'. (b)Find the equation of the tangent to the folium of Descartesx3+y3 =6xy at point (3,3). up down return end

  6. Orthogonal:Two curves are calledOrthogonal,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. ExampleThe equations xy=c (c0) represents a family of hyperbolas. And the The equations x2-y2=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. 2.7Higher derivatives Derivativef' (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: Similarly f ''' (x)=[f '' (x)] ' is called the third derivative of f(x), and 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 definef(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), ExampleIf f(x)= , find f(n)(x). ExampleIf y=x4-3x2+6x+9, find y', y' ', y' ' ', y(4). ExampleIf f(x)=sinx, g(x)=cosx, find f(n)(x) and g(n)(x) . ExampleFind y ' ' , if x4+y3 =x-y . up down return end

  9. 2.8Related rates (omitted)

  10. 2.9Differentials, Linear and Quadratic Approximations Definition:Let x=x-x0, f(x) =f(x)-f(x0). If there exists a constant A(x0) which is independent of x and x such that f(x)=A(x0) x+B(x, x0) where B(x, x0) satisfies . Then A x is called differential of f(x) at x0. Generally A x is denoted by df(x)|x=x0= A(x0) x. Replacing x0 by x, the differential is denoted by df(x) and df(x)= A(x) x. up down return end

  11. Corollary: If the differential of f(x) is df(x)= A(x) x, then f(x) is differentiable and A(x)=f'(x). Proof: From the definition, 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. y y=f(x) Q R dy P S dx=x x o x t As x=dx is very small,y=dy ,i.e., f(t)-f(x)f '(x) t. Example (a)Find dy, if y=x3+5x4. (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 up down return end

  13. 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. Example Use differentials to find an approximate (65)1/3 . From definition of the differential, we can easily get up down return end

  14. ExampleFind 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: Suppose f(x) is a function which the second derivative f' '(a) exists. P(x)=A+Bx+Cx2 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. 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. Corolary: Suppose P(x)=A+Bx+Cx2 is the Quadratic approximation to f(x) near a. Then P(x)=f(a)+f '(a)(x-a)+ f' '(a)(x-a)2 /2. up down return end

  16. ExampleFind the quadratic approximation to f(x)=cosx near 0. up down return end

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

  18. Suppose f(x) is defined on [a,b], f '(x) does not value 0. Let x0[a,b], f(a)f(b)<0. And x1=x0- , x2=x1- . Keeping repeating the process (xn=xn-1- ), we obtain a sequence of approximations x1 , x2 ,..., xn ,...... If , then r is the root of the equation f(x)=0. 2.10Newton’s method(to be omitted) The method is to give a way to get a approximation to a root of an equation. up down return end

  19. ExampleStarting with x1=2, find the third approximation x3 to the root of the equation x3-2x-5=0. up down return end

  20. 2.1Derivatives 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 derivativeof a function f at a number a, denoted by f´(a), is if this limit exists. Then we have: up down return end

  22. ExampleFind the derivative of the function y=x2-8x+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. ExampleFind the equation of the tangent line of the function y=x2-3x+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. ExampleIf f(x)=(x-1)1/2, find the derivative of f . State the domain of f´(x). Example Find 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)=Dxf(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. ExampleWhere 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) There are several cases a function fails to be differentiable (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; (3) the points at which the curve has a vertical tangent line, such as, f(x)=x1/3, at x=0. up down return end

  27. 2.2Differentiation 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)=xn, where n is a positive integer, then f´(x)= nxn-1, xn =nxn-1. ExampleIf f(x)=x100, find f´(x). up down return end

  29. 3)Theorem Suppose c is a constant and f´(x) and g´(x) exist.Then (a) (cf(x))´ exists and (cf(x))´=cf´(x); (b) (f(x)+g(x))´exists and (f(x)+g(x)´=f´(x)+g´(x); (c) (f(x)-g(x))´exists and (f(x)-g(x)´=f´(x)-g´(x). Example If f(x)= x50 +x100, find f´(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) . ExampleIf f(x)= (2x5)(3x10), find f´(x). up down return end

  31. x2+2x-5 x3-6 ExampleIf f(x)= , find f´(x). 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. up down return end

  32. 2). The power rule (general version) If f(x)=xn, where n is any real number, then f´(x)= nxn-1, ,i.e., xn =nxn-1. Example If f(x)=x, find f´(x). If g(x)= x1/2, g´(x)=? ExampleDifferentiate the function f(t)=(1-t)t1/3. Table of differentiation formulas (in paper 119) up down return end

  33. 2.3Rate of change in the Economics C C(x2)-C(x1) C(x1+x)-C(x1) = = x x2-x1 x SupposeC(x) is the total cost that a company incurs in producing x units of certain commodity. The function C is called a cost function. If the number of items produced increased from x1 to x2, the additional cost is C= C(x2)-C(x1), and the average of change of the cost is up down return end

  34. 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). The limit of this quantity as x0, is called the marginal cost by economist. 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. 2.4Derivatives of trigonometric functions y D B A  o C x -1 (1) Theorem Proof: suppose OP=1 and (0, /2). So we will show Notice that 0<|BC|<arcAB (2) Corollary up down return end

  36. (3) Theorem y D B A  o C x -1 Proof: Notice that Area of OAB<Area of sector OAB<Area of OAD. (4) Corollary up down return end

  37. ExampleFind up down return end

  38. (5) Theorem Example Differentiate y=xsinx. (6) Theorem ExampleDifferentiate y=tanx. Corollary (tanx)'=sec2x up down return end

  39. ExampleDifferentiate y=cotx. Corollary (cotx)'= - csc2x ExampleDifferentiate f(x)= up down return end

  40. 2.5Chain 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). Proof: Let u=g(x+x)-g(x) y=f(u+u)-f(u) up down return end

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

  42. ExampleFind F '(x) if F(x)=(1+x2)3/4. up down return end

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