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Higher Order Derivatives Implicit Differentiation Mathematics 53 Institute of Mathematics (UP Diliman) Institute of Mathematics (UP Diliman) Higher Order, Implicit Differentiation Mathematics 53 1 / 19

M53 Lec2.3A Higher Order Derivatives Implicit Differentiation

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  • Higher Order Derivatives

    Implicit Differentiation

    Mathematics 53

    Institute of Mathematics (UP Diliman)

    Institute of Mathematics (UP Diliman) Higher Order, Implicit Differentiation Mathematics 53 1 / 19

  • For today

    1 Higher Order Derivatives

    2 Implicit Differentiation

    Institute of Mathematics (UP Diliman) Higher Order, Implicit Differentiation Mathematics 53 2 / 19

  • Higher Order Derivatives

    If f is differentiable, then the derivative of f is called the second derivative of fand is denoted f .

    We can continue to obtain the third derivative, f , the fourth derivative, f (4),and other higher derivatives.

    DefinitionThe n-th derivative of the function f , denoted f (n), is the derivative of the(n 1)-th derivative of f , that is,

    f (n)(x) = limx0

    f (n1)(x+ x) f (n1)(x)x

    Institute of Mathematics (UP Diliman) Higher Order, Implicit Differentiation Mathematics 53 4 / 19

  • Higher Order Derivatives

    Remarks

    1 The n in f (n) is called the order of the derivative.

    2 The derivative of a function f is also called the first derivative of f .

    3 The function f is sometimes written as f (0)(x).

    4 Other notations: Dxn[ f (x)] ,dnydxn

    ,dn

    dxn[ f (x)] , y(n)

    Institute of Mathematics (UP Diliman) Higher Order, Implicit Differentiation Mathematics 53 5 / 19

  • Higher Order Derivatives

    Example

    Find f (n)(x) for all n N where f (x) = x6 x4 3x3 + 2x2 4.

    Solution. We differentiate repeatedly and obtain

    f (x) = 6x5 4x3 9x2 + 4x,f (x) = 30x4 12x2 18x+ 4,f (x) = 120x3 24x 18,f (4)(x) = 360x2 24,f (5)(x) = 720x,

    f (6)(x) = 720,

    f (n)(x) = 0 for all n 7.

    Institute of Mathematics (UP Diliman) Higher Order, Implicit Differentiation Mathematics 53 6 / 19

  • Higher Order Derivatives

    Example

    Find f (4)(x) if f (x) =2x 3.

    Solution.f (x) = 1

    2(2x 3) 12 (2) = (2x 3) 12

    f (x) = 12(2x 3) 32 (2) = (2x 3) 32

    f (x) = 32(2x 3) 52 (2) = 3 (2x 3) 52

    f (4)(x) = 152(2x 3) 72 (2) = 15 (2x 3) 72

    Institute of Mathematics (UP Diliman) Higher Order, Implicit Differentiation Mathematics 53 7 / 19

  • Equations in two variables are used to define a function explicitly or implicitly.

    y = x2 1 defines f (x) = x2 1 (explicitly).

    y2 = x+ 1 defines two functions of x: (implicitly)

    f1(x) =x+ 1

    f2(x) = x+ 1.

    Institute of Mathematics (UP Diliman) Higher Order, Implicit Differentiation Mathematics 53 9 / 19

  • Consider: If x3 2xy 3y 6 = 0, how do we find dydx

    ?

    SolutionFrom x3 2xy 3y 6 = 0, we can define y explicitly in terms of x as

    y =x3 62x+ 3

    .

    Thus,dydx

    =(2x+ 3)(3x2) (x3 6)(2)

    (2x+ 3)2=

    4x3 + 9x2 + 12(2x+ 3)2

    .

    Institute of Mathematics (UP Diliman) Higher Order, Implicit Differentiation Mathematics 53 10 / 19

  • Implicit Differentiation

    Question: Suppose we have x4y3 7xy = 7 or tan(x2 2xy) = y.How do we find

    dydx

    ?

    Institute of Mathematics (UP Diliman) Higher Order, Implicit Differentiation Mathematics 53 11 / 19

  • Implicit Differentiation

    To finddydx

    using implicit differentiation, we

    1 think of the variable y as a differentiable function of the variable x,

    2 differentiate both sides of the equation, using the chain rule where necessary,

    and

    3 solve fordydx

    .

    Institute of Mathematics (UP Diliman) Higher Order, Implicit Differentiation Mathematics 53 12 / 19

  • Implicit Differentiation

    Example

    Finddydx

    if x3 2xy 3y 6 = 0.

    Solution.

    Dx[x3 2xy 3y 6] = Dx[0]Dx[x3] Dx[2xy] Dx[3y] Dx[6] = Dx[0]3x2 2

    (1 y+ x dy

    dx

    ) 3 dy

    dx 0 = 0

    3x2 2y 2x dydx 3 dy

    dx= 0

    2x dydx

    + 3 dydx

    = 3x2 2ydydx

    (2x+ 3) = 3x2 2y

    Institute of Mathematics (UP Diliman) Higher Order, Implicit Differentiation Mathematics 53 13 / 19

  • Implicit Differentiation

    dydx

    =3x2 2y2x+ 3

    =3x2 2 x362x+3

    2x+ 3

    =3x2 (2x+ 3) 2(x3 6)

    (2x+ 3)2

    =4x3 + 9x2 + 12(2x+ 3)2

    .

    Institute of Mathematics (UP Diliman) Higher Order, Implicit Differentiation Mathematics 53 14 / 19

  • Implicit Differentiation

    Example

    Finddydx

    if x4y3 7xy = 7.

    Solution.

    Dx[x4y3 7xy] = Dx[7]4x3y3 + 3x4y2 dy

    dx 7(y+ x dy

    dx

    )= 0

    dydx

    =4x3y3 + 7y3x4y2 7x .

    Institute of Mathematics (UP Diliman) Higher Order, Implicit Differentiation Mathematics 53 15 / 19

  • Implicit Differentiation

    Example

    Finddydx

    if tan(x2 2xy) = y.

    Solution.

    Dx[tan(x2 2xy)] = Dx[y]sec2(x2 2xy)

    [2x 2

    (y+ x dy

    dx

    )]=

    dydx

    Thus,

    dydx

    =2x sec2(x2 2xy) 2y sec2(x2 2xy)

    1+ 2x sec2(x2 2xy) .

    Institute of Mathematics (UP Diliman) Higher Order, Implicit Differentiation Mathematics 53 16 / 19

  • Higher Order Derivatives

    Example

    Determined2ydx2

    if xy2 = y 2.

    Solution.

    Dx[xy2] = Dx[y 2]y2 + x 2y dy

    dx=

    dydx

    dydx

    =y2

    1 2xy

    Institute of Mathematics (UP Diliman) Higher Order, Implicit Differentiation Mathematics 53 17 / 19

  • We havedydx

    =y2

    1 2xy .

    = d2ydx2

    =(1 2xy)(2y dydx ) (y2)[2(y+ x dydx )]

    (1 2xy)2

    =(1 2xy)

    [2y y212xy

    ] (y2)

    [2(y+ x y212xy

    )](1 2xy)2 .

    Institute of Mathematics (UP Diliman) Higher Order, Implicit Differentiation Mathematics 53 18 / 19

  • Exercises

    1 Determine D3x [ cos(4x) ].

    2 Finddydx

    if

    sin(x+ y) = xy2 2x3.3 Find the equation of the tangent line to the graph of

    3x cos y = 1+ x2 +2y+ 1 at P(1, 0).

    4 Find the fourth derivative ofx6 1x2

    .

    Institute of Mathematics (UP Diliman) Higher Order, Implicit Differentiation Mathematics 53 19 / 19

    Higher Order DerivativesImplicit Differentiation