> f:=x->x^2+7*x+5; > g:=x^3-5*x+8;Here are some examples that show how

> D(f)(x); > diff(g,x); > diff(f(x),x);See what happens when the function or expression notations are used incorrectly.

> 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. Again, pay attention to the difference between expressions and functions.

> diff(g,x,x); > diff(g,x,x,x); > (D@@2)(f)(x); > (D@@3)(f)(x);If you want to evaluate the higher derivative at a specific value of

> (D@@2)(f)(2); > subs(x=3,diff(g,x));Suppose you wanted to find the equation of the tangent line to the graph of

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

> f:=x^2*y^2+y^3=0; > implicitdiff(f,y,x);where

> 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);Suppose you want to find the equation of the line tangent to the graph of defined implicitly. For instance, find the equation of the line tangent to at the point . You can use the point-slope form of a line in implicit form to get the equation of the tangent line. The Maple commands below show how this can be done.

> f:=x^2*y^2+y^3=0; > m:=subs({x=1,y=-1},implicitdiff(f,y,x)); > tanline := y-(-1)=m*(x-1); > implicitplot({f,tanline},x=0..2,y=-3..0);

Sometimes you want the value of a derivative, but first have to find the coordinates of the point. More than likely, you will have to use the `solve` or `fsolve` command for this. However, to get the `fsolve` command to give you the solution you want, you often have to specify a range for the variable. Plotting the graph of a relation can be a big help in this task. For instance, if you wanted the slope of as in the previous example at and is negative, but the value is not given, then you would first need to solve for by substituting the value into and then solve for . See how this is done below.

> f:=x^2*y^2+y^3=0; > solve(subs(x=1,f),y);

- Evaluate the fourth derivative of
, at
using function notation and
- the
**D**command. - the
**diff**command.

- the
- Given the implicit relation ,
- Plot the relation over the interval and .
- Find all values when .
- Find the equation of the line tangent to the graph at the point where and is negative.
- Plot the relation and the tangent line on the same graph over the interval and .

- Consider the graph of the ellipse defined implicitly by the equation . Find the equations of the tangent lines to this curve at the two points where it intersects the x-axis. Supply a plot of the ellipse and the two tangent lines on one graph.

2006-11-14