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Higher order partial derivatives

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Intros
Lessons
  1. High Order Partial Derivatives Overview:
  2. 2nd Order Partial Derivatives
    • 4 types of 2nd order partial derivatives
    • fxx,fxy,fyy,fyxf_{xx}, f_{xy}, f_{yy}, f_{yx}
    • An example
  3. Higher Order Partial Derivatives
    • Can go higher than 2nd order
    • fxxx,fxxy,fxxxxxf_{xxx}, f_{xxy}, f_{xxxxx}
    • An example
  4. 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
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Examples
Lessons
  1. Finding 2nd Order Partial Derivatives
    Find all the second order derivatives for the following function

    f(x,y)=x3y−4xy3+ln⁡(x2) f(x,y) = x^3 y - \sqrt{4xy^3} + \ln (x^2)

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

      f(x,y)=ex2y3−sin⁡(x2+y3)f(x,y) = e^{x^2y^3} - \sin (x^2 + y^3)

      1. Finding Higher Order Partial Derivatives
        Given w=est+sin⁡(s2)w=e^{st}+ \sin (s^2), find wsssttw_{ssstt}
        1. Given f(x,y,z)=4(xyz)3f(x,y,z)={^4}\sqrt{(xyz)^3} , find d4fdy2dx2 \frac{d^4f}{dy^2dx^2}
          1. Verifying Clairaut's Theorem
            Verify Clairaut's Theorem for the given function

            u(x,y)=ln⁡(x2−y)u(x,y) = \ln (x^2 - y)

            1. Verify Clairaut's Theorem for the given function

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

              Topic Notes
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              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:
              fxx→f_{xx} \to derivative in respect to xx 2 times
              fyy→f_{yy} \to derivative in respect to yy 2 times
              fxy→f_{xy} \to derivative in respect to xx first, and then respect to yy
              fyx→f_{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)