Higher order partial derivatives

Higher order partial derivatives

Lessons

Notes:

2nd Order Partial Derivatives
Since we can have higher order derivatives on a one variable function, we can also have this for multi-variable functions. We will specifically look at 2nd order partial derivatives here. For 2nd order partial derivatives, there are 4 types:

fxx=ddx(dfdx)=d2fdx2f_{xx} = \frac{d}{dx}(\frac{df}{dx}) = \frac{d^2f}{dx^2}
fxy=ddy(dfdx)=d2fdydxf_{xy} = \frac{d}{dy}(\frac{df}{dx}) = \frac{d^2f}{dydx}
fyy=ddy(dfdy)=d2fdy2f_{yy} = \frac{d}{dy}(\frac{df}{dy}) = \frac{d^2f}{dy^2}
fyx=ddx(dfdy)=d2fdxdyf_{yx} = \frac{d}{dx}(\frac{df}{dy}) = \frac{d^2f}{dxdy}

Where:
fxxf_{xx} \to derivative in respect to xx 2 times
fyyf_{yy} \to derivative in respect to yy 2 times
fxyf_{xy} \to derivative in respect to xx first, and then respect to yy
fyxf_{yx} \to derivative in respect to yy first, and then respect to xx

Higher Order Partial Derivatives
Of course, we can have even higher order partial derivatives. For example, we can have:

fxxx=ddx(d2fdx2)=d3fdx3f_{xxx} = \frac{d}{dx} (\frac{d^2f}{dx^2}) = \frac{d^3f}{dx^3}
fxxy=ddy(d2fdx2)=d3fdydx2f_{xxy} = \frac{d}{dy} (\frac{d^2f}{dx^2}) = \frac{d^3f}{dydx^2}
fxxxxx=ddx(d4fdx4)=d5fdx5f_{xxxxx} = \frac{d}{dx} (\frac{d^4f}{dx^4}) = \frac{d^5f}{dx^5}

We cannot list them all here because there is an infinite amount of higher order partial derivatives.

Clairaut's Theorem Suppose that ff is defined on a disk DD that contains the point (a,b)(a,b). If the functions fxyf_{xy} and fyxf_{yx} are continuous on this disk, then

fxy(a,b)=fyx(a,b)f_{xy}(a,b) = f_{yx}(a,b)

  • Introduction
    High Order Partial Derivatives Overview:
    a)
    2nd Order Partial Derivatives
    • 4 types of 2nd order partial derivatives
    • fxx,fxy,fyy,fyxf_{xx}, f_{xy}, f_{yy}, f_{yx}
    • An example

    b)
    Higher Order Partial Derivatives
    • Can go higher than 2nd order
    • fxxx,fxxy,fxxxxxf_{xxx}, f_{xxy}, f_{xxxxx}
    • An example

    c)
    Clairaut's Theorem
    • Two of the 2nd order partial derivatives are equal!
    • fxy(a,b)=fyx(a,b)f_{xy}(a,b) = f_{yx}(a,b)
    • An example to show they are equal


  • 1.
    Finding 2nd Order Partial Derivatives
    Find all the second order derivatives for the following function

    f(x,y)=x3y4xy3+ln(x2) f(x,y) = x^3 y - \sqrt{4xy^3} + \ln (x^2)


  • 2.
    Find fxxf_{xx} and fxy f_{xy} for the following function

    f(x,y)=ex2y3sin(x2+y3)f(x,y) = e^{x^2y^3} - \sin (x^2 + y^3)


  • 3.
    Finding Higher Order Partial Derivatives
    Given w=est+sin(s2)w=e^{st}+ \sin (s^2), find wsssttw_{ssstt}

  • 4.
    Given f(x,y,z)=4(xyz)3f(x,y,z)={^4}\sqrt{(xyz)^3} , find d4fdy2dx2 \frac{d^4f}{dy^2dx^2}

  • 5.
    Verifying Clairaut's Theorem
    Verify Clairaut's Theorem for the given function

    u(x,y)=ln(x2y)u(x,y) = \ln (x^2 - y)


  • 6.
    Verify Clairaut's Theorem for the given function

    f(x,y)=xtanxy+exy+x5f(x,y) = x \tan \frac{x}{y} + e^{xy} + x^5