- The Limit Definition of the Derivative
- The Maple D and diff commands
- The Equation of a Tangent Line
- Exercises

It can be interpreted geometrically as the slope of the tangent line to the graph of at a point and functionally as the instantaneous rate of change of at . You can use the definition and the Maple limit command to compute derivatives directly, as shown below. You can also compute derivatives using Maple's

> limit((f(x+h)-f(x))/h,h=0);The example below shows how to use the limit definition of derivative to find with Maple.

> f := x -> x^2+3*x+5; > limit((f(1+h)-f(1))/h,h=0);

- The
`D`command acts on a function. - The
`diff`command acts on an expression or a function 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, be careful with the parentheses. It is one of the only commands in Maple where the gets its own parentheses.

> f:=x->x^2; > D(f)(x);Finding the derivative at a specific value is easy. (Again be careful of the parentheses.)

> D(f)(2);

The `D` operator **CANNOT** be used on expressions. To differentiate expressions, you need to use the `diff` command. Here is an example.

> p:=3*x+2; > diff(p,x);Remember the

> pprime:=diff(p,x); > subs(x=2,pprime);Another option is to embed the commands.

>subs(x=2,diff(p,x));

> tly:= D(f)(5)*(x-5)+f(5); > plot([f(x),tly],x=0..10);

- For the function
- A)
- Find the derivative of the functionusing the limit definition of the derivative, the
`diff`command and then the`D`command (Make sure the output from all three are the same. You may need to use the`simplify`command.) - B)
- And then use all three methods to find the slope of at . (Again make sure the output from all three are the same. You may need to use the
`evalf`command.

- Find the equation of the line tangent to the graph of the function
at . Include a plot of the function and the tangent line on the same graph over the interval
.
- For the function
, find all points on the graph of where the tangent line is horizontal.
- A)
- Plot the function and state how many points you are looking for.
- B)
- Find the values.
- C)
- Find the values. Then state in text the points where there is a horizontal tangent line. State your answers in decimal form. (Remember in the text sentence to use two decimals, rounding correctly.)

2011-08-17