, you can do this in Maple by first defining the difference quotient and the computing the limit. The following example shows how to compute the derivative of and the evaluate the derivative at .

> f:=x-> x^3; > quot := (f(x+h)-f(x))/h; > der:=limit(quot,h=0); > subs(x=-2,der);or if you don't need to see that the derivative of is and you just want to evaluate the derivative at , then this could be done all in one Maple command as in the following example:

> f:=x->x^3; > limit((f(-2+h)-f(-2))/h,h=0));

- The
`D`operator acts on a function to produce the derivative of that function. - The
`diff`command acts on an expression and differentiates that expression with respect to a variable specified by the user.

When you use the `D` operator to compute the derivative of a
function, the result is also a function, as shown below.

> D(f);If you provide a label, then you get a function you can use later in the session,

> df := D(f);However, this is usually not necessary. See the examples below.

If you want to evaluate the derivative at a specific value of or
just get the expression for the derivative, you can use the following
forms of the `D` operator.

> D(f)(2); > D(f)(x);This last form is the one to use for plotting, as shown below.

> plot(D(f)(x),x=-2..2);

Suppose you want to find the equaton of the line tangent to the graph of at the point . This can be done in Maple using the point slope form of a line as shown below.

> tanline := D(f)(5)*(x-5)+f(5);

The `D` operator cannot be used on expressions, for example
trying to use it to differentiate the expresssion we defined above
results in an error.

> D(p);If you recall that Maple uses

> D(f(x));

To differentiate expressions, you need to use the `diff`
command. Here is an example.

> diff(p,x);The

> diff(f(x),x);Note, however, that the result of the

> der := diff(p,x); > subs(x=Pi/2,der);

- Show that the derivative of the function is by using the limit definition of derivative. Use a variable name to label the derivative. Then evaluate the derivative at . Then calculate the derivative at again by plugging directly into the difference quotient.
- Use a variable name and expression notation to define
,
- Compute the derivative and label the derivative using a variable name.
- Evaluate the derivative at , , and .
- Plot the expression over the interval . Using this plot, can you explain why the expression was not differentiable at one of the values given above?

- Find the equation of the line tangent to the graph of the function
at . When calculating the derivative at a point, use the command. Include a plot of the function and the tangent line on the same graph over the interval
.
- For the same function
,
- Plot over the interval and state how many horizontal tangent lines to the graph there are.
- Plot the derivative of over the same interval. Explain how this supports your answer above.
- Using the
`fsolve`command along with labels, find each value where a horizontal tangent line is located. Find the corresponding values by plugging each value back into the function. State in text all points on the graph of where the tangent line is horizontal.

2007-09-13