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Section 3: Partial Derivatives

Instructions

  • First, you should watch the concepts videos below explaining the topics in the section. 
  • Second, you should attempt to solve the exercises and then watch the videos explaining the exercises. 
  • Last, you should attempt to answer the self-assessment questions to determine how well you learned the material.
  • When you have finished the material below, you can start on the next section or return to the main several variable calculus page.

Concepts

  • The definition the partial derivative of \(f(x,y)\) with respect to \(x\) and \(y\)
  • The geometric interpretation of the partial derivative
  • Higher order partial derivatives and Clairaut’s Theorem
 



If you would like to see more videos on this topic, click the following link and see the related videos. Note the related videos at the link are not required viewing.
Partial Derivatives Conceptual V1.2

Exercises


Directions: You should attempt to solve the problems first and then watch the video to see the solution. 
  1. If \(f(x,y)=4-x^2-2y^2\), find \(f_x(1,1)\) and \(f_y(1,1)\) and interpret them as slopes.

    \( f_x(1,1)=-2\)
    \(f_y(1,1)=-4\)
    See the video for the interpretation.


    If you would like to see more videos on this topic, click the following link and see the related videos. Note the related videos at the link are not required viewing.
    Partial Derivatives Exercise V1


  2. If \(f(x,y)=y\sin(-x)+2y\), find \(\left . \dfrac{\partial f}{\partial x}\right|_{(\pi,0)}\) and 
    \(\left . \dfrac{\partial f}{\partial y}\right|_{(\pi,0)}.\)

    \(\left . \dfrac{\partial f}{\partial x}\right|_{(\pi,0)}=0\)  
    \(\left . \dfrac{\partial f}{\partial y}\right|_{(\pi,0)}=2\)


    If you would like to see more videos on this topic, click the following link and see the related videos. Note the related videos at the link are not required viewing.
    Partial Derivatives Exercise V2


  3. Find \(f_y(x,y)\) if \(f(x,y)=e^{\tan(xy^3)}.\)

    \(f_y(x,y)=\sec^2\left(xy^3\right)\left(3xy^2\right)e^{\tan\left(xy^3\right)}\)


    If you would like to see more videos on this topic, click the following link and see the related videos. Note the related videos at the link are not required viewing.
    Partial Derivatives Exercise V3


  4. Find all higher order partial derivatives of \(f(x,y)=xe^y+y^2e^{6x}.\)

    \(f_{xx}(x,y)=36y^2e^{6x}\)
    \(f_{xy}(x,y)=e^y+12ye^{6x}\)
    \(f_{yy}(x,y)=xe^y+2e^{6x}\)
    \(f_{yx}(x,y)=e^y+12ye^{6x}\)


    If you would like to see more videos on this topic, click the following link and see the related videos. Note the related videos at the link are not required viewing.
    Partial Derivatives Exercise V4


 

Self-Assessment Questions


Directions: The following questions are an assessment of your understanding of the material above. If you are not sure of the answers, you may need to rewatch the videos. 
  1. What is the difference between the partial derivative of \(z=f(x,y)\) with respect to \(x\) and then with respect to \(y\)?
  2. Give an example where you must use the product rule to find \(f_x(x,y)\).
  3. Sketch a graph of a surface that illustrates the geometric interpretation of \(f_x(1, 1)\) and \(f_y(1, 1)\).
  4. Is it always true that \(f_{xy}(x,y)=f_{yx}(x,y)\)?