** Next:** About this document ...
**Up:** Labs and Projects for
** Previous:** Labs and Projects for

Maple knows how to take many derivatives. Its main commands for doing this are **D** and **diff**. **D** is designed to differentiate functions, whereas **diff** is for differentiating expressions. However, if proper notation is used, **diff** can also be used with functions. To review the difference between a function and an expression, check the two examples below. The *f* statement defines a function, the *g* statement defines an expression.

> f:=x->x^2+7*x+5;

> g:=x^3-5*x+8;Here are some examples that show how

> D(sin);

> D(sin)(x);Also, check these two.

> D(f);

> D(f)(x);Carefully consider these.

> diff(g,x);

> diff(f(x),x);See what happens with these.

> diff(g(x),x);

> diff(f,x);After the last four examples, you should be convinced that proper notation is very important in doing derivatives in Maple.

Maple can also do higher derivatives. Check these commands.

> diff(g,x2);

> diff(f(x),x$2);The

> (D@@2)(f)(x);

Just as Maple can plot several functions with a single command, it can differentiate more than one function at a time.

> diff({sin(x)^2,sin(x^2)},x);However, note that while Maple does what is asked, it rearranges the order of the output. So, be careful.

More information on **diff** and **D** can be obtained through Maple help screens.

> (f@f)(x);

> simplify(");

> h:=x->x^3+x^2-3*x+4;

> p:=x->8*x+6;

> (f@h@p)(x);

> simplify(");

> (p@h@f)(x);

> simplify(");Are and the same?

The **implicitdiff** command can be used to find derivatives of implicitly defined functions. The syntax is as follows

> implicitdiff(f,y,x);where

> f:=x^2*y^2+y^3;

> implicitdiff(f,y,x);

> g:=x^2+y^3=1;

> implicitdiff(g,y,x);Second derivatives can also be taken with

> implicitdiff(g,y,x,x);Maple also has a command for plotting implicitly defined functions. It is in the package

> with(plots);

> implicitplot(x^2-y^2=1,x=-3..3,y=-3..3);

- 1.
- Type and execute all commands given in the discussions above.
- 2.
- A point (
*x*,_{0}*f*(*x*)) on the graph of_{0}*y*=*f*(*x*) is known as a local minimum (maximum) point if for all*x*in some open interval containing*x*._{0}- (a)
- Plot . What appears to be true about the slope of the tangent line at each local minimum or maximum point?
- (b)
- Use the observation you have just made to find the
*x*-coordinate of each local minimum or maximum point. Note that the`numer`command could be helpful here.

- 3.
- Find the point on the graph of that is closest to the point (2.1, 0.8).
- 4.
- Consider the function .
- (a)
- In order to take the derivative of the given function by the Chain Rule, it is necessary to describe the function as the composite of several functions. Prepare for applying the Chain Rule by finding appropriate functions such that is the given function. Use Maple to show that your breakdown is correct.
- (b)
- Attach a handwritten page showing how the derivative is taken by means of the Chain Rule.
- (c)
- Use Maple to check your answer in (b).

- 5.
- Consider a function
*y*implicitly defined by the equation*y*- 2^{4}*x*+^{2}y^{2}*x*= 17.^{3}- (a)
- Attach a handwritten page on which you find .
- (b)
- Use Maple to check your answer in (a).
- (c)
- Find an equation of the line tangent to the graph at the point (2,3).

12/1/1998