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- 1. Partial Derivatives
- 2. Partial DerivativesLet p = (a, b, c) be a point on the surface z = f(x, y).
- 3. Partial DerivativesLet p = (a, b, c) be a point on the surface z = f(x, y).The plane y = b intersects the surface
- 4. Partial DerivativesLet p = (a, b, c) be a point on the surface z = f(x, y).The plane y = b intersects the surface z y=b (a, b, c) y x
- 5. Partial DerivativesLet p = (a, b, c) be a point on the surface z = f(x, y).The plane y = b intersects the surface at the curvez = f(x, b) y=b z z y=b z = f(x, b) (a, c) (a, b, c) x y x
- 6. Partial DerivativesLet p = (a, b, c) be a point on the surface z = f(x, y).The plane y = b intersects the surface at the curvez = f(x, b) which is a y–trace that contains p = (a, b, c). y=b z z y=b z = f(x, b) (a, c) (a, b, c) x y x
- 7. Partial DerivativesLet p = (a, b, c) be a point on the surface z = f(x, y).The plane y = b intersects the surface at the curvez = f(x, b) which is a y–trace that contains p = (a, b, c).The slope at p on this trace in the y = b plane, y=b z z y=b z = f(x, b) (a, c) (a, b, c) x y x
- 8. Partial DerivativesLet p = (a, b, c) be a point on the surface z = f(x, y).The plane y = b intersects the surface at the curvez = f(x, b) which is a y–trace that contains p = (a, b, c).The slope at p on this trace in the y = b plane, is calledthe partial derivative with respect to x. y=b z z y=b z = f(x, b) (a, c) (a, b, c) x y x
- 9. Partial DerivativesLet p = (a, b, c) be a point on the surface z = f(x, y).The plane y = b intersects the surface at the curvez = f(x, b) which is a y–trace that contains p = (a, b, c).The slope at p on this trace in the y = b plane, is calledthe partial derivative with respect to x. Specifically,The partial derivative of z = f(x, y) with respect to x is df dz dx = dx y=b z z y=b z = f(x, b) (a, c) (a, b, c) x y x d z gives the slopes of y traces. dx
- 10. Partial DerivativesLet p = (a, b, c) be a point on the surface z = f(x, y).The plane y = b intersects the surface at the curvez = f(x, b) which is a y–trace that contains p = (a, b, c).The slope at p on this trace in the y = b plane, is calledthe partial derivative with respect to x. Specifically,The partial derivative of z = f(x, y) with respect to x is df dz f(x + h, y) – f(x, y) , if it exists. d x = d x = h 0 lim h y=b z z y=b z = f(x, b) (a, c) (a, b, c) x y x d z gives the slopes of y traces. dx
- 11. Partial Derivatives dzAlgebraically, to find d x , we treat y as a constantand apply the derivative with respect to x.
- 12. Partial Derivatives dzAlgebraically, to find d x , we treat y as a constantand apply the derivative with respect to x.Example A. Find dx dza. z = x + y b. z = xyc. z = x2y d. z = xy2 ye. z = x y f. z = x
- 13. Partial Derivatives dzAlgebraically, to find d x , we treat y as a constantand apply the derivative with respect to x.Example A. Find dx dza. z = x + y dz = 1 b. z = xy dx because y = 0.c. z = x2y d. z = xy2 ye. z = x y f. z = x
- 14. Partial Derivatives dzAlgebraically, to find d x , we treat y as a constantand apply the derivative with respect to x.Example A. Find dx dza. z = x + y dz = 1 b. z = xy dz = y dx dx because y = 0. because (cx) = c.c. z = x2y d. z = xy2 ye. z = x y f. z = x
- 15. Partial Derivatives dzAlgebraically, to find d x , we treat y as a constantand apply the derivative with respect to x.Example A. Find dx dza. z = x + y dz = 1 b. z = xy dz = y dx dx because y = 0. because (cx) = c. dzc. z = x y 2 dx = 2xy d. z = xy2 because (xn) = nxn–1. x ye. z = y f. z = x
- 16. Partial Derivatives dzAlgebraically, to find d x , we treat y as a constantand apply the derivative with respect to x.Example A. Find dx dza. z = x + y dz = 1 b. z = xy dz = y dx dx because y = 0. because (cx) = c. dz dzc. z = x y 2 dx = 2xy d. z = xy 2 dx = y2 because (xn) = nxn–1. because (cx) = c. x ye. z = y f. z = x
- 17. Partial Derivatives dzAlgebraically, to find d x , we treat y as a constantand apply the derivative with respect to x.Example A. Find dx dza. z = x + y dz = 1 b. z = xy dz = y dx dx because y = 0. because (cx) = c. dz dzc. z = x y 2 dx = 2xy d. z = xy 2 dx = y2 because (xn) = nxn–1. because (cx) = c. x dz 1 ye. z = y dx = y f. z = x because (cx) = c.
- 18. Partial Derivatives dzAlgebraically, to find d x , we treat y as a constantand apply the derivative with respect to x.Example A. Find dx dza. z = x + y dz = 1 b. z = xy dz = y dx dx because y = 0. because (cx) = c. dz dzc. z = x y 2 dx = 2xy d. z = xy 2 dx = y2 because (xn) = nxn–1. because (cx) = c. x dz 1 y dze. z = y dx = y f. z = dx = –yx –2 x because (cx) = c. The Quotient Rule
- 19. Partial Derivatives dzAlgebraically, to find d x , we treat y as a constantand apply the derivative with respect to x.Example A. Find dx dza. z = x + y dz = 1 b. z = xy dz = y dx dx because y = 0. because (cx) = c. dz dzc. z = x y 2 dx = 2xy d. z = xy 2 dx = y2 because (xn) = nxn–1. because (cx) = c. x dz 1 y dze. z = y dx = y f. z = dx = –yx –2 x because (cx) = c. The Quotient RuleThe plane y = b is parallel to the x–axis, so the partialdz/dx is also referred to as the partial derivative in thex–direction (as in the compass direction).
- 20. Partial DerivativesExample B. Find an equation for the tangent line forat (2, 3, 6) on the surface of z = f(x, y) = √49 – x2 – y2in the direction of the x-axis.
- 21. Partial DerivativesExample B. Find an equation for the tangent line forat (2, 3, 6) on the surface of z = f(x, y) = √49 – x2 – y2in the direction of the x-axis. Reminder: the graph of z = √49 – x2 – y2 is the top half of the hemisphere with radius 7.
- 22. Partial DerivativesExample B. Find an equation for the tangent line forat (2, 3, 6) on the surface of z = f(x, y) = √49 – x2 – y2in the direction of the x-axis. z y=3 (2,3,6) y x
- 23. Partial DerivativesExample B. Find an equation for the tangent line forat (2, 3, 6) on the surface of z = f(x, y) = √49 – x2 – y2in the direction of the x-axis. z y=3 (2,3,6) y x
- 24. Partial DerivativesExample B. Find an equation for the tangent line forat (2, 3, 6) on the surface of z = f(x, y) = √49 – x2 – y2in the direction of the x-axis.We need df/ dx|(2,3). z y=3 (2,3,6) y x
- 25. Partial DerivativesExample B. Find an equation for the tangent line forat (2, 3, 6) on the surface of z = f(x, y) = √49 – x2 – y2in the direction of the x-axis.We need df/ dx|(2,3). (i.e. the slope this tangent) z y=3 (2,3,6) y x
- 26. Partial DerivativesExample B. Find an equation for the tangent line forat (2, 3, 6) on the surface of z = f(x, y) = √49 – x2 – y2in the direction of the x-axis.We need df/ dx|(2,3). df/dx = –x/√49 – x2 – y2 , z y=3 (2,3,6) y x
- 27. Partial DerivativesExample B. Find an equation for the tangent line forat (2, 3, 6) on the surface of z = f(x, y) = √49 – x2 – y2in the direction of the x-axis.We need df/ dx|(2,3). df/dx = –x/√49 – x2 – y2 , at (2, 3),we get df / dx|(2,3) = –2/6 = –1/3 z y=3 (2,3,6) y x
- 28. Partial DerivativesExample B. Find an equation for the tangent line forat (2, 3, 6) on the surface of z = f(x, y) = √49 – x2 – y2in the direction of the x-axis.We need df/ dx|(2,3). df/dx = –x/√49 – x2 – y2 , at (2, 3),we get df / dx|(2,3) = –2/6 = –1/3 = slope as shown. z y=3 z y=3 1 (2,3,6) –1/3 x y x
- 29. Partial DerivativesExample B. Find an equation for the tangent line forat (2, 3, 6) on the surface of z = f(x, y) = √49 – x2 – y2in the direction of the x-axis.We need df/ dx|(2,3). df/dx = –x/√49 – x2 – y2 , at (2, 3),we get df / dx|(2,3) = –2/6 = –1/3 = slope as shown. So adirections vector for the tangent is v = <1, 0, –1/3> z y=3 z y=3 1 (2,3,6) –1/3 x y x
- 30. Partial DerivativesExample B. Find an equation for the tangent line forat (2, 3, 6) on the surface of z = f(x, y) = √49 – x2 – y2in the direction of the x-axis.We need df/ dx|(2,3). df/dx = –x/√49 – x2 – y2 , at (2, 3),we get df / dx|(2,3) = –2/6 = –1/3 = slope as shown. So adirections vector for the tangent is v = <1, 0, –1/3> andan equation for the tangent is L = <t + 2, 3, –t/3 + 6>. z y=3 z y=3 1 (2,3,6) –1/3 x y x
- 31. Partial DerivativesThe geometry of the partial derivative with respect to yis similar to partial derivative with respect to x.
- 32. Partial DerivativesThe geometry of the partial derivative with respect to yis similar to partial derivative with respect to x.Let (a, b, c) be a point on the surface of z = f(x, y),the plane x = a intersects the surface at a curve that isthe graph of the equation z = f(a, y), in the plane x = a. x=a (a,b,c) y x
- 33. Partial DerivativesThe geometry of the partial derivative with respect to yis similar to partial derivative with respect to x.Let (a, b, c) be a point on the surface of z = f(x, y),the plane x = a intersects the surface at a curve that isthe graph of the equation z = f(a, y), in the plane x = a. dz dy |P = slope of the tangent of z = f(a, y) in the plane x = a at the point y= b . x=a (a,b,c) y x
- 34. Partial DerivativesThe geometry of the partial derivative with respect to yis similar to partial derivative with respect to x.Let (a, b, c) be a point on the surface of z = f(x, y),the plane x = a intersects the surface at a curve that isthe graph of the equation z = f(a, y), in the plane x = a. dz dy |P = slope of the tangent of z = f(a, y) in the plane x = a at the point y= b . x=a x=a z (a,b,c) (b, c) z = f(a,y) y y x
- 35. Partial DerivativesThe geometry of the partial derivative with respect to yis similar to partial derivative with respect to x.Let (a, b, c) be a point on the surface of z = f(x, y),the plane x = a intersects the surface at a curve that isthe graph of the equation z = f(a, y), in the plane x = a. dz dy |P = slope of the tangent of z = f(a, y) in the plane x = a at the point y= b . x=a x=a z (a,b,c) (b, c) z = f(a,y) y y x d z gives the slopes of x traces. dy
- 36. Partial Derivatives dzTo find d y , treat x as a constant in the formula.
- 37. Partial Derivatives dzTo find d y , treat x as a constant in the formula. dz dzExample C. a. Find dx , dy if z = cos(x2y).
- 38. Partial Derivatives dzTo find d y , treat x as a constant in the formula. dz dzExample C. a. Find dx , dy if z = cos(x2y).We need to use the chain–rule for both partials.
- 39. Partial Derivatives dzTo find d y , treat x as a constant in the formula. dz dzExample C. a. Find dx , dy if z = cos(x2y).We need to use the chain–rule for both partials. dz d(x2y) dx = –sin(x2y)* dx
- 40. Partial Derivatives dzTo find d y , treat x as a constant in the formula. dz dzExample C. a. Find dx , dy if z = cos(x2y).We need to use the chain–rule for both partials. dz d(x2y) dx = –sin(x2y)* dx = –sin(x2y)*2xy
- 41. Partial Derivatives dzTo find d y , treat x as a constant in the formula. dz dzExample C. a. Find dx , dy if z = cos(x2y).We need to use the chain–rule for both partials. dz d(x2y) dz d(x2y) dy = –sin(x y)* dy 2 dx = –sin(x2y)* dx = –sin(x2y)*2xy
- 42. Partial Derivatives dzTo find d y , treat x as a constant in the formula. dz dzExample C. a. Find dx , dy if z = cos(x2y).We need to use the chain–rule for both partials. dz d(x2y) dz d(x2y) dy = –sin(x y)* dy 2 dx = –sin(x2y)* dx = –sin(x y)*2xy 2 = –sin(x2y)*x2
- 43. Partial Derivatives dzTo find d y , treat x as a constant in the formula. dz dzExample C. a. Find dx , dy if z = cos(x2y).We need to use the chain–rule for both partials. dz d(x2y) dz d(x2y) dy = –sin(x y)* dy 2 dx = –sin(x2y)* dx = –sin(x y)*2xy 2 = –sin(x2y)*x2b. Find dz , dz at the point P = (1, π/2, 0) dx dy
- 44. Partial Derivatives dzTo find d y , treat x as a constant in the formula. dz dzExample C. a. Find dx , dy if z = cos(x2y).We need to use the chain–rule for both partials. dz d(x2y) dz d(x2y) dy = –sin(x y)* dy 2 dx = –sin(x2y)* dx = –sin(x y)*2xy 2 = –sin(x2y)*x2b. Find dz , dz at the point P = (1, π/2, 0) dx dy dz dx |(1,π/2, 0)
- 45. Partial Derivatives dzTo find d y , treat x as a constant in the formula. dz dzExample C. a. Find dx , dy if z = cos(x2y).We need to use the chain–rule for both partials. dz d(x2y) dz d(x2y) dy = –sin(x y)* dy 2 dx = –sin(x2y)* dx = –sin(x y)*2xy 2 = –sin(x2y)*x2b. Find dz , dz at the point P = (1, π/2, 0) dx dy dz dx |(1,π/2, 0) = –sin(x2y)*2xy|(1,π/2,0) = –π
- 46. Partial Derivatives dzTo find d y , treat x as a constant in the formula. dz dzExample C. a. Find dx , dy if z = cos(x2y).We need to use the chain–rule for both partials. dz d(x2y) dz d(x2y) dy = –sin(x y)* dy 2 dx = –sin(x2y)* dx = –sin(x y)*2xy 2 = –sin(x2y)*x2b. Find dz , dz at the point P = (1, π/2, 0) dx dy dz dz dx |(1,π/2, 0) dy |(1,π/2, 0) = –sin(x2y)*2xy|(1,π/2,0) = –π
- 47. Partial Derivatives dzTo find d y , treat x as a constant in the formula. dz dzExample C. a. Find dx , dy if z = cos(x2y).We need to use the chain–rule for both partials. dz d(x2y) dz d(x2y) dy = –sin(x y)* dy 2 dx = –sin(x2y)* dx = –sin(x y)*2xy 2 = –sin(x2y)*x2b. Find dz , dz at the point P = (1, π/2, 0) dx dy dz dz dx |(1,π/2, 0) dy |(1,π/2, 0) = –sin(x2y)*2xy|(1,π/2,0) = –sin(x2y)*x2|(1,π/2,0) = –π = –1
- 48. Partial DerivativesGiven a function of one variable, we say the function is"differentiable" at a point P if the derivative exists at P.
- 49. Partial DerivativesGiven a function of one variable, we say the function is"differentiable" at a point P if the derivative exists at P. f(x) exists means the curve is smooth, not a corner, at P, so its tangent is well defined.
- 50. Partial DerivativesGiven a function of one variable, we say the function is"differentiable" at a point P if the derivative exists at P. f(x) exists means the curve is smooth, not a corner, at P, so its tangent is well defined. p f(x) exists
- 51. Partial DerivativesGiven a function of one variable, we say the function is"differentiable" at a point P if the derivative exists at P. f(x) exists means the curve is smooth, not a corner, at P, so its tangent is well defined. p p f(x) exists f(x) doesnt exist
- 52. Partial DerivativesGiven a function of one variable, we say the function is"differentiable" at a point P if the derivative exists at P. f(x) exists means the curve is smooth, not a corner, at P, so its tangent is well defined. p p f(x) exists f(x) doesnt existHowever, for a function f(x, y) of two variables(or more), the existence of the partial derivativesdoesnt mean the surface z = f(x, y) is smooth,it may have creases (i.e. folds) even
- 53. Partial DerivativesGiven a function of one variable, we say the function is"differentiable" at a point P if the derivative exists at P. f(x) exists means the curve is smooth, not a corner, at P, so its tangent is well defined. p p f(x) exists f(x) doesnt existHowever, for a function f(x, y) of two variables(or more), the existence of the partial derivativesdoesnt mean the surface z = f(x, y) is smooth,it may have creases (i.e. folds) evenif the partials exist.
- 54. Partial DerivativesGiven a function of one variable, we say the function is"differentiable" at a point P if the derivative exists at P. f(x) exists means the curve is smooth, not a corner, at P, so its tangent is well defined. p p f(x) exists f(x) doesnt existHowever, for a function f(x, y) of two variables(or more), the existence of the partial derivativesdoesnt mean the surface z = f(x, y) is smooth,it may have creases (i.e. folds) even P yif the partials exist. For example,the surface shown here has dx dz |P = dz |P = 0, x dybut the surface is folded, i.e. not smooth at P.
- 55. Partial DerivativesIf we impose the conditions that 1. both partials withrespect to x and y exist in a small circle with P as thecenter, z p y x (a, b)
- 56. Partial DerivativesIf we impose the conditions that 1. both partials withrespect to x and y exist in a small circle with P as thecenter, and 2. that both partial derivatives arecontinuous in this circle, z p y x (a, b)
- 57. Partial DerivativesIf we impose the conditions that 1. both partials withrespect to x and y exist in a small circle with P as thecenter, and 2. that both partial derivatives arecontinuous in this circle, then the surface is smooth(no crease) at P and it has a well defined tangentplane at P as shown. A smooth point P and its tangent plane. z p y x (a, b)
- 58. Partial DerivativesIf we impose the conditions that 1. both partials withrespect to x and y exist in a small circle with P as thecenter, and 2. that both partial derivatives arecontinuous in this circle, then the surface is smooth(no crease) at P and it has a well defined tangentplane at P as shown. A smooth point P and its tangent plane.Hence we say z = f(x,y) is z"differentiable" or "smooth" at P p dzif both partials dz , dy exist, dx yand are continuous in a xneighborhood of P. (a, b)
- 59. Partial Derivatives If we impose the conditions that 1. both partials with respect to x and y exist in a small circle with P as the center, and 2. that both partial derivatives are continuous in this circle, then the surface is smooth (no crease) at P and it has a well defined tangent plane at P as shown. A smooth point P and its tangent plane. Hence we say z = f(x,y) is z "differentiable" or "smooth" at P p dz if both partials dz , dy exist, dx y and are continuous in a x neighborhood of P. (a, b)For any elementary functions z = f(x, y), if a small circlethat centered at P is in the domain, that all the partialsexist in this circle then the surface is smooth at P.
- 60. Partial DerivativesWe also write df as fx and df as dx dy fy .
- 61. Partial DerivativesWe also write df as fx and df as dx dyIts easy to talk about highery . f derivatives with thisnotation. So, fxx means to take the partial derivativewith respect to x twice, fxy means to to take thederivative with respect to x first, then with respect to y.
- 62. Partial DerivativesWe also write df as fx and df as dx dyIts easy to talk about highery . f derivatives with thisnotation. So, fxx means to take the partial derivativewith respect to x twice, fxy means to to take thederivative with respect to x first, then2 with respect to y.In the quotient form fxx is written as d f dxdx
- 63. Partial DerivativesWe also write df as fx and df as dx dyIts easy to talk about highery . f derivatives with thisnotation. So, fxx means to take the partial derivativewith respect to x twice, fxy means to to take thederivative with respect to x first, then2 with respect to y.In the quotient form fxx is written as d f dxdxIn the quotient form fxy is written as d2f dydx
- 64. Partial DerivativesWe also write df as fx and df as dx dyIts easy to talk about highery . f derivatives with thisnotation. So, fxx means to take the partial derivativewith respect to x twice, fxy means to to take thederivative with respect to x first, then2 with respect to y.In the quotient form fxx is written as d f dxdxIn the quotient form fxy is written as d2f dydx Note the reversed orders in these notations
- 65. Partial DerivativesWe also write df as fx and df as dx dyIts easy to talk about highery . f derivatives with thisnotation. So, fxx means to take the partial derivativewith respect to x twice, fxy means to to take thederivative with respect to x first, then2 with respect to y.In the quotient form fxx is written as d f dxdxIn the quotient form fxy is written as d2f dydx Note the reversed orders in these notationsHence fxyy = d f and that f = d f . 3 3 dydydx yxx dxdxdy
- 66. Partial DerivativesWe also write df as fx and df as dx dyIts easy to talk about highery . f derivatives with thisnotation. So, fxx means to take the partial derivativewith respect to x twice, fxy means to to take thederivative with respect to x first, then2 with respect to y.In the quotient form fxx is written as d f dxdxIn the quotient form fxy is written as d2f dydx Note the reversed orders in these notationsHence fxyy = d f and that f = d f . 3 3 dydydx yxx dxdxdyIn most cases, the order of the partial is notimportant.
- 67. Partial DerivativesWe also write df as fx and df as dx dyIts easy to talk about highery . f derivatives with thisnotation. So, fxx means to take the partial derivativewith respect to x twice, fxy means to to take thederivative with respect to x first, then2 with respect to y.In the quotient form fxx is written as d f dxdxIn the quotient form fxy is written as d2f dydx Note the reversed orders in these notationsHence fxyy = d f and that f = d f . 3 3 dydydx yxx dxdxdyIn most cases, the order of the partial is notimportant.Theorem: If fxy and fyx exist and are continuous in a
- 68. Partial DerivativesAgain, for elementary functions, we have f xy = fyxfor all points inside the domain (may not be truefor points on the boundary of the domain.)
- 69. Partial DerivativesAgain, for elementary functions, we have f xy = fyxfor all points inside the domain (may not be truefor points on the boundary of the domain.)Example D. Find fxx, fyy, fxy, and fyx if f(x, y) = x2y – 2x3y.
- 70. Partial DerivativesAgain, for elementary functions, we have f xy = fyxfor all points inside the domain (may not be truefor points on the boundary of the domain.)Example D. Find fxx, fyy, fxy, and fyx if f(x, y) = x2y – 2x3y.The first order partials are:fx = 2xy – 6x2y,
- 71. Partial DerivativesAgain, for elementary functions, we have f xy = fyxfor all points inside the domain (may not be truefor points on the boundary of the domain.)Example D. Find fxx, fyy, fxy, and fyx if f(x, y) = x2y – 2x3y.The first order partials are:fx = 2xy – 6x2y, fy = x2 – 2x3
- 72. Partial DerivativesAgain, for elementary functions, we have f xy = fyxfor all points inside the domain (may not be truefor points on the boundary of the domain.)Example D. Find fxx, fyy, fxy, and fyx if f(x, y) = x2y – 2x3y.The first order partials are:fx = 2xy – 6x2y, fy = x2 – 2x3Hence,fxx = 2y – 12xy,
- 73. Partial DerivativesAgain, for elementary functions, we have f xy = fyxfor all points inside the domain (may not be truefor points on the boundary of the domain.)Example D. Find fxx, fyy, fxy, and fyx if f(x, y) = x2y – 2x3y.The first order partials are:fx = 2xy – 6x2y, fy = x2 – 2x3Hence,fxx = 2y – 12xy, fyy = 0
- 74. Partial DerivativesAgain, for elementary functions, we have f xy = fyxfor all points inside the domain (may not be truefor points on the boundary of the domain.)Example D. Find fxx, fyy, fxy, and fyx if f(x, y) = x2y – 2x3y.The first order partials are:fx = 2xy – 6x2y, fy = x2 – 2x3Hence,fxx = 2y – 12xy, fyy = 0fxy = 2x – 6x2 = fyx

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