Calculus I using LiveMath

Chapter 3

Implicit Differentiation

The function y = f(x) = x^2 defines y as a differentiable function of x, and you should have no trouble finding its derivative. The graph of the function along with its derivative (red line) is shown in the following graphic:

On the other hand, y^2 = x is not a function, but a relation. This is because, for each value of x there is more than one value for y. In the graph below, the arbitrary chosen lines a and b cross the plot twice, violating the vertical line property of functions. To solve for the derivative, you would have to split the equation into two separate functions, as described below.

You could turn this into a function by defining two separate branches:

You could find the derivative of each separate function, but to solve for the slope of a chosen point on the line you would have to pick the correct function first. You would do this by looking at the quadrant the point was in.

Another solution would be to use the method discussed next.

A function is explicit when you can place it in the form: y=f (x). For example, the following is an equation in explicit form:

(all the x's and no y's are on the right side of the equation).

To find its derivative would be an easy task using the rules you have studied in this chapter. The same equation in implicit form looks like:

You would normally solve for y before you could easily find the derivative. In this example solving for y is not dificult.

The problem comes when it is difficult to explicitly solve an equation for y. You may want to look at the following steps to help you solve these problems:

• Differentiate both sides of the implicit equation with respect to the independent variable. In the example above, this is x.
• Move and collect terms so that dy/dx is on one side of the equation.
• Solve for dy/dx

The following example will show you how to use LiveMath to solve these types of equations.

Example 3.16 Implicit Differentiation

 Find the equation of the line tangent to the equation x^2+y^2 = 25 at x =3.

• Enter the equation, select it whole by clicking on the equal sign, and envoke the Select-in manipulation.
  
• Apply the Derivative Op by clicking on the Pallette. This applies the Op to both sides at the same time. After you do this, type an x defining the independent variable of the equation.
  
• Simplify
  
  Note what happened here (remember that the derivative of a constant, 25 here, is always equal to zero, More ):
  
• Isolate the Derivative Op (turn off Auto-Casing)
  
• Now you have the equation for the slope. In other words:
  
• To find the slope at x = 3 (the original question), you need to find out what y is equal to. You do this by substituting x = 3 into the original equation and solving for y.
  
  
• All you have to do now is substitute these values into the slope equation for the answer.
  

  

• You have the slope of the line tangent to the original function (which, by the way, you should recognize as a circle). But, what is the equation of that line? All you need do now is fill in the values you have found into the equation for a line using the point-slope form. Below, this is shown for you.
  
• Isolate y and Expand for the final answer.
  
• Adding the line to a graph of the circle gives the following plot
  
  Notice how the line has a negative slope as indicated by the negative value of m, you determined eariler.

LiveMath has a graphing facility that allows you to plot implicit graphs atuomatically. The next example shows you how.

3.17 Implicit Differentiation - Graphing

• Input the same equation used in the last example.
  
• Select and choose the Implicit graph under the Graph 2D menu. The following Graph Theory will appear after you choose True Proportions in the graph details. 3.3 NOTE:
  
• Add the tangent line just like you did in the previous example. The graph below results.
  

The following demonstration uses LiveMath to draw two tangent lines on the circle while displaying the value of the slopes of both (the blue line merely locates the point of tangency for you).

Circle Slopes LiveMath Internet

• As you work this demonstration, ask yourself; "Why does a have to be between -5 and 5 ?"