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MA 1024, Partial derivatives


The purpose of this lab is to acquaint you with using Maple to compute partial derivatives.

Getting Started

To assist you, there is a worksheet associated with this lab that contains examples. You can copy that worksheet to your home directory with the following command, which must be run in a terminal window, not in Maple.

cp /math/calclab/MA1024/ My_Documents

Another way to access the getting started worksheet is to go to your computer's Start menu and choose run. In the run field type:

when you hit enter, you can then choose MA1024 and then choose the worksheet
Remember to immediately save it in your home directory. Once you've copied and saved the worksheet, read through the background on the internet and the background of the worksheet before starting the exercises.


For a function $f(x)$ of a single real variable, the derivative $f'(x)$ gives information on whether the graph of $f$ is increasing or decreasing. Finding where the derivative is zero was important in finding extreme values. For a function $F(x,y)$ of two (or more) variables, the situation is more complicated.

Partial derivatives

A differentiable function, $F(x,y)$, of two variables has two partial derivatives: $\partial F /\partial x$ and $\partial F /\partial
y$. As you have learned in class, computing partial derivatives is very much like computing regular derivatives. The main difference is that when you are computing $\partial F /\partial x$, you must treat the variable $y$ as if it was a constant and vice-versa when computing $\partial F /\partial

The Maple commands for computing partial derivatives are D and diff. The Getting Started worksheet has examples of how to use these commands to compute partial derivatives.


  1. Compute the three distinct second order partial derivatives of

    \begin{displaymath}f(x,y)=\frac{\ln(x^2+1)}{y^2+1} \end{displaymath}

    at the point $(-1,1)$ using the diff command and then again using the D command.

  2. Given the function

    \begin{displaymath}k(x,y)=(\sin(x)+\sin(y))(x^3-7xy^2+2x^2y) \end{displaymath}

    do the following:
    Plot the function and the plane $x=4$ on the same graph. Use plotting ranges $-5 \leq x \leq 5$, $-5 \leq y \leq 5$, $-700 \leq z \leq 700$.
    Find the derivative of $k$ in the $x=4$ plane.
    Find the equation of the tangent line at the point $y=-3$ in the $x=4$ plane.
    Graph the two-dimensional intersection of the plane $x=4$ and $k$ along with the tangent line at $y=-3$. Be sure to use $y$ and $z$ ranges that are consistent with your ranges in part a.
    Does your two-dimensional graph look like the intersection from your tree-dimensional graph?

  3. Given:

    \begin{displaymath}j(x,y) = \frac{x+y}{2+x^2+y^2} \end{displaymath}

    Find all points on the graph where the tangent plane is horizontal.
    Find the equation of each horizontal plane and plot them on the same graph as the function $j(x,y)$. Use plotting ranges $-5 \leq x \leq 5$, $-5 \leq y \leq 5$.

next up previous
Next: About this document ... Up: lab_template Previous: lab_template
Dina J. Solitro-Rassias