The **implicitdiff** command can be used to find derivatives of
implicitly defined functions. Suppose we wanted to use implicit
differentiation to find
for the relation

Then we first define our relation and give it a label for later use.

> f:=x^2*y^2+y^3=0;Note that the

> implicitdiff(f,y,x);

The result of the command is the derivative,
. The first argument is the relation
that you want to differentiate implicitly. We were careful to use an
equation for this argument, but if you just give an expression for
this argument, Maple assumes you want to set this expression equal to
zero before differentiating. The second argument to the
`implicitdiff` command is where you tell Maple what the
**dependent variable** is. The remaining argument is to
specifying the derivative you want.

To compute numerical values of derivatives obtained by implicit differentiation, you have to use the subs command. For example, to find the value of at the point you could use the following command.

> subs([x=1,y=-1],implicitdiff(f,y,x));

> with(plots):The

> f := x^2*y^2+y^3 = 0;g := x^2*sin(y) = 1; > implicitplot([f,g],x=-10..10,y=-10..0);You can add options to the plot command to make the picture more clear. Execute the following commands to see the improvements.

> implicitplot([f,g],x=-10..10,y=-10..0,numpoints=10000); > implicitplot([f,g],x=-10..10,y=-10..0,numpoints=10000,color=[``Aqua'', ''Magenta'']); > implicitplot([f,g],x=-10..10,y=-10..0,numpoints=10000,color=[``Aqua'', ''Magenta''],scaling=constrained);To find where the graphs intersect you can use the

> a := fsolve({f, g}, {x = -5 .. 0, y = -5 .. 0}); > b := fsolve({f, g}, {x = -5 .. 0, y = -5 .. 0}); > c := fsolve({f, g}, {x = 0 .. 5, y = -5 .. 0}); > d := fsolve({f, g}, {x = 0 .. 5, y = -10 .. 5});You could then use these points to find the slopes of

> afslope := subs(a, implicitdiff(f, y, x)); > agslope := evalf(subs(a, implicitdiff(g, y, x))); > afline := afslope*(x-a[1])+a[2]; > agline := agslope*(x-b[1])+b[2]; > implicitplot([g, f, afline, agline], x = -10 .. 10, y = -10 .. 5, numpoints = 10000, color = ["Aqua", "Magenta", "DarkOliveGreen", "Blue"], scaling = constrained);

- Enter the following equations as expressions in Maple:
and the vertical line .
- A)
- Plot the two equations experimenting with the domain and range and the options to get a good picture of the intersections.
- B)
- Using the
`solve`command find the y values of the interesction points. - C)
- Find the slope of the first equation at the intersection point with the positive y value.
- C)
- State the intersection point and its slope. Then looking at the graph give at least two reason that the slope value seems to make sense.

- Given the relation
,
- A)
- Plot the graph of the relation over the interval and .
- B)
- Using
`fsolve`commands, find the points where the equation has horizontal tangents. Remember you are solving the derivative and the equation simultaneously so you will need to use curly brackets. State your points in text. - C)
- Plot the equation and the horizontal tangent lines. (Remember, these are horizontal lines so the equations should be simple.)

2011-08-21