Subsections

• Purpose
• Background
• Tangent line to an explicit function at a point
  • Implicit Differentiation
  
  
• Tangent line to an implicitly defined relation at a point
• Exercises

Implicit Differentiation

Purpose

The purpose of this lab is to give you experience using Maple to compute derivatives to find tangent lines to an expression or function both explicitly and implicitly.

Background

Tangent line to an explicit function at a point

Suppose you want to find the equaton of the line tangent to the graph of $f(x)$ at the point $x=5$. Using the point slope form of a line, this can be found and plotted in Maple as shown below.

> f:=x->x^3;
> tanline := D(f)(3)*(x-5)+f(3);
> plot([f(x),tanline],x=0..6);

Implicit Differentiation

The implicitdiff command can be used to find derivatives of implicitly defined functions. Suppose we wanted to use implicit differentiation to find $$\frac{dy}{dx}$$ for the relation

$$x^2y^2+y^3=0$$

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

> f:=x^2+y^2=25;

A plot may be necessary to determine how many y values there are for a given x value. For instance, at $x=3$, we can show that $y=-4$ and $y=4$:

> with(plots):
> implicitplot(f,x=-6..6,y=-6..6);
> solve(subs(x=3,f),y)

The syntax of the implicitdiff command is shown by the following example.

> der:=implicitdiff(f,y,x);

The result of the command is the implicit derivative, $$\frac{dy}{dx}$$. The syntax of this command is very similar to that of the diff command. The first argument is always 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. That is, by putting y here, we were saying that we were thinking of this relation as defining $y(x)$ and not $x(y)$. The remaining arguments to implicitdiff are for specifying the order of the derivative you want.

Second derivatives can also be computed with implicitdiff. The following command computes $$\frac{d^2y}{dx^2}$$.

> der2:=implicitdiff(f,y,x,x);

To compute numerical values of derivatives obtained by implicit differentiation, you have to use the subs command. For example, to find the value of $$\frac{dy}{dx}$$ at the point $(3,-4)$ you could use the following command.

> m:=subs({x=3,y=-4},implicitdiff(f,y,x));

Tangent line to an implicitly defined relation at a point

Suppose you wanted to find the equation of the tangent line to the graph of $f$ at the point $(1,-1)$. You may want to label the output to $$\frac{dy}{dx}$$ at the point $(1,-1)$ as $m$ for slope and then you can use the point-slope form of a line to get the equation of the tangent line. The Maple commands below show how this can be done.

> m := subs({x=3,y=-4},implicitdiff(f,y,x));
> tanline := y-(-4)=m*(x-3);
> with(plots):
> implicitplot({f,tanline},x=-6..6,y=-6..6);

We can also use implicit differentiation to find higher order derivatives. For example, we may want to determine the concavity of a graph by evaluating the second derivative at a point as shown below:

> der2 := implicitdiff(f,y,x,x)
> subs({x=3,y=-4},der2)

Exercises

1. Given the relation $18x^2y^3+12xy^2+x-48y=1$,

   A)

   Define the equation as an expression $f$ and plot the graph of the relation over the interval $-5 \leq x \leq 5$ and $-2 \leq y \leq 2$. How many $y$ values are there at $x=1$? Find them using the solve command.

   B)

   Find the slope of the graph at the point where $x=1$ and $y$ is negative. Label the slope as m.

   C)

   Find the equation of the line tangent to the graph of the given relation at the point where $x=1$ and $y$ is negative. Be sure to label the tangent line as an implicit relation using the point-slope form of a line. Supply a plot of the relation and the tangent line over the intervals $0 \leq x \leq 5$ and $-4 \leq y \leq 0$.

2. Enter the following equation as expression $g$ in Maple: $$x^2y-\frac{4}{5}xy+\frac{4}{25}y+y^2=2$$.

   A)

   Plot the relation $g$ using the plotting intervals $-5 \leq x \leq 8$ and $-5 \leq y \leq 3$. Looking at the graph, how many points have the x value $$\frac{-3}{5}$$? Using the solve command find the y value(s).

   B)

   Find the slope of the tangent at each point and name them $m1$, $m2$, ...

   C)

   Find the equation of the line(s) tangent to $g$ at each point and label them. Plot the relation and the tangent line(s) on the same graph using the same plotting intervals given above.

   D)

   Looking at your graph in part A, is the concavity positive or negative at each point whose x value is $$\frac{-3}{5}$$. Find $$\frac{d^2y}{dx^2}$$ of each of the points to confirm.