Shared07 - Implicit Differentiation Assignment / Implicit Differentiation Notes.sagewsOpen in CoCalc

This material was developed by Aaron Tresham at the University of Hawaii at Hilo and is

licensed under a Creative Commons Attribution-ShareAlike 4.0 International License.

licensed under a Creative Commons Attribution-ShareAlike 4.0 International License.

- Intro to Sage
- Tangent Lines
- Differentiation

In this lab we will explore implicit functions (of two variables), including their graphs, derivatives, and tangent lines.

An example of an implicit function is given by the equation . This equation provides an implicit relation between and . Compare this to the equation , which gives explicitly in terms of .

Graphing an implicit function is fairly simple in Sage using the implicit_plot command. This command requires three arguments: an equation (using double equal sign), a plot range for the first variable, and a plot range for the second variable. I will add the optional "axes=true" and "frame=false" so that axes will be plotted instead of a frame.

Graph (circle of radius 5 centered at the origin).

Graph .

Now that we can graph these functions, we want to compute the derivative of with respect to . This assumes that is a function of , so we need to tell Sage to assume this as well:

Now we can take the derivative.

Find if .

First, we take the derivative of the whole equation, then we'll solve for .

2*y(x)*diff(y(x), x) + 2*x == 0

The diff(y(x),x) is the derivative .

The curvy-looking "d" you get when you use show is the symbol for a partial derivative (you'll learn about those in Calc 3). Since this is Calc 1, you should just think of those as a regular "d."

Now we can solve for the derivative:

[diff(y(x), x) == -x/y(x)]

[]

This tells us that . [Note: Sage is treating as a function of , so it uses function notation . We usually write just .]

Find when .

[diff(y(x), x) == -1/2*(3*x^2 + y(x)^2 + 2*y(x))/((x - 1)*y(x) + x)]

[]

So .

Now that we can find the derivative of an implicit function, we can also find tangent lines.

Recall that the line tangent to a function at the point has equation .

Find an equation for the line tangent to the circle given by at the point .

Above we found . So the slope of the tangent line at is .

Thus, an equation for the tangent line is .

Let's graph the implicit function and the tangent line.

Find an equation for the tangent line to the graph of at the point .

We found the derivative above: .

Now we need to substitute and .

I will copy and paste this derivative from the calculation above, and then I will replace x with and y(x) with .

-9/8

Now that we have the slope, we can find an equation of the tangent line: .

Let's check our answer by graphing:

Here is one final example that puts all the pieces together.

Consider

Find the derivative, find the tangent line at , and graph the curve and tangent line.

First, we find the derivative.

[diff(y(x), x) == -32*(x^3 + x*y(x)^2 + 6*x*y(x))/(32*x^2*y(x) + 32*y(x)^3 + 96*x^2 - 81*y(x)^2)]

[]

Now we define "a" and "b," copy and paste the derivative, and replace x with a and y(x) with b.

22.4547850705957

Next we define the tangent line, using the answer above for the slope.

22.4547850705957*x - 11.0956306531714

Finally, we plot the original function and the tangent line (remember to "reset" y using %var y before graphing).

Sage can also plot an implicit function of three variables. We won't need this for our assignment, but here are a few examples.

[Note: you can make it bigger or smaller with the mouse wheel; click and drag to rotate]

Sphere of radius 5:

3D rendering not yet implemented

This one is a little more interesting:

3D rendering not yet implemented

3D rendering not yet implemented