Note:All example commands given in this write-up and the output from them should be included in your lab report.

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,x$2);

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

> (D@@2)(f)(x);More information on

> (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.
- Find an equation of the line tangent to the graph of at the point (2,0). (Note that Maple has a
**sqrt**function.) - 2.
- Find the third derivative of evaluated at .
- 3.
- Consider the function . Find the points at which the graph of this function has a horizontal tangent line. (The
**numer**command could be useful here.) Use**plot**to graph the function. What geometric property seems to be indicated by a horizontal tangent line? - 4.
- Consider the function
- (a)
- In order to take the derivative of the given function by using the Chain Rule, it is necessary to picture the function as the composite of several functions. Open a text section and explain how that is done for this problem.
- (b)
- Use Maple to take the composite of the functions you found in (a). Make sure this composite really gives the original function.
- (c)
- Open a text section and explain how the derivative is found by means of the Chain Rule.
- (d)
- Check your answer in (c) by using Maple to get the derivative.

- 5.
- Consider a function
*y*that is implicitly defined by*y*- 3^{5}*x*+^{2}y*x*= -1.^{3}- (a)
- Attach a handwritten page on which you find .
- (b)
- Use Maple to check the accuracy of your work.
- (c)
- Find an equation of the line tangent to the graph of the function at the point (1,1).

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