At the end of this session you will be able to
"IMPLICITLY DEFINED" FUNCTIONS
When you write y = f(t) for the dependent variable of a function f, we say that the function is "explicitly defined" and we have an explicit function. In y = f(t) we are explicitly giving the rule for the function. Up to now, all of your functions in this course have been explicitly defined.
These are examples of "explicitly defined" functions:
y = sin t
y = x2 - 3x + 5
f(x) = cos (4t3 -5t)
The equations 1. x2 + y2 = 64 and 2. y5 + sin (xy) = 0 define relations, but do not define functions. On the other hand, 3. y = x3 + 2x - 3 does define a function.
Each can be written in the form F(x,y) = 0 with simple algebra. For example, they are respectively
- F(x,y) = x2 + y2 - 64 = 0
- F(x,y) = y5 + sin (xy) = 0
- F(x,y) = x3 + 2x - 3 - y = 0
When we have a relation or a function written in the form F(x,y) = 0, we say it is written implicitly. However, in our three examples written in the form F(x,y) = 0, we would not know they are functions unless we were told that they are or unless we could solve for y.
In Example 1. x2 + y2 - 64 = 0, you could solve for y by using algebra such as the quadratic formula, but you would obtain two answers because there are two square roots. Try it. Thus, we have two functions.
In Example 2. y5 + sin (xy) = 0, you can solve for y at all. Therefore, you do not know if you have a function.
In Example 3. x3 + 2x - 3 - y = 0, you could easily solve for y, expressing the function explicitly.
In all three examples 1. and 2. above, you could have a function if you had certain restrictions placed on the ranges of y = f(x). We will encounter this situations where F(x,y) = 0 in calculus does represent a function. Then we say that the function is "implicitly defined".
In summary, if we know that we have a function f, then y = f(t) explicitly defines the function and F(t,y) = 0 implicitly defines the function.
DIFFERENTIATE IMPLICITLY
As we have seen, sometimes we have a function "implicitly defined" and we cannot express the function explicitly by solving for y. If we do have a function, then we will still want to know what the derivative is.
So how do we find he derivative? On page 207 of your textbook, you have the three steps of implicit differentiation outlined in the upper left-hand corner of the page.
Step 1. should read "Differentiate both sides of the equation with respect to x using the Chain Rule, treating y as a differentiable function of x."
What your textbook means in Step 2. "Collect the terms with dy/dx on one side of the equation." is to use elementary algebra to have all terms with dy/dx on one side of the equation and everything else on the other side.
Both Examples 3 and Examples 4 on pages 207 and 208 are excellent in illustrating the process of implicit differentiation.
CALCULATE HIGH ORDER DERIVATIVES IMPLICITLY
As you look at Example 5 on page 209 of your textbook, you see that we still go through the same page 207 steps to find the second order derivative using implicit differentiation. The only additional step is at the end where you substitute the first derivative into the second order derivative.
COMPLETE APPLICATIONS INVOLVING DERIVATIVES OF IMPLICIT FUNCTIONS
We know that the geometric interpretation of a derivative is the
Also, we have learned in previous lessons how to use the slope-intercept form of the equation of a straight line to write the equation of the tangent line.
In this lesson we have learned about a new line, called the normal line. As pictured in Figure 3.40 on page 207 and Figure 3.41 on page 208, the normal line is perpendicular to the tangent line. Even with implicit differentiation, we find the equation of the tangent line and normal line as you would with explicit functions, except you use implicit differentiation to find the derivative. Example 4 on page 208 is an outstanding illustration.
Now you can go to the Session 11 assignments to continue your assignment.
At the end of this session you will be able to
DERIVATIVE POWER RULE FOR RATIONAL POWERS
When you studied the bottom of page 209 and the top of page 210, you noticed that the Power Rule for Rational Powers is exactly the same as the Power Rule for Integer Powers. If you have forgotten what a rational number is, then you need to review pages AP-12 and AP-13 about real number subsets.
The proof of the Power Rule for Rational Numbers utilizes the Chain Rule and implicit differentiation. Otherwise, all knowledge that is needed in the proof is the laws for exponents for integers.
APPLY THE DERIVATIVE POWER RULE FOR RATIONAL POWERS
In applying Power Rule for Rational Numbers, you saw in Example 5 on page 209 that you differentiate just like you do when you have integers for exponents.
Now you can go to the Session 11 assignments to continue your assignment.
At the end of this session you will be able to
RELATED RATES
In related rate problems, the rates are instantaneous rates of changes, which means they might be velocity and acceleration problems. Always the independent variable is time in a related rate problem. The reason they are called related rate problems is because we have variables that are related and, thus, the derivatives with respect to time will be related.
APPLICATIONS INVOLVING RELATED RATES
Thus in related rate problems we have variables that are related and we have time as the independent variable. If variables are related by time, then their derivatives with respect to time are related.
Therefore, you need to determine the relationship between variables. Subsequently, use implicit differentiation to differentiate with respect to time. The Related Rate Problem Strategy on page 215 is your guide to solving the applications. Use the steps just as outlined there.
In Step 5, differentiate implicitly with respect to t. Your book does not say to differentiate implicitly, but assumes you automatically know that. Well, not everybody realizes that. So add differentiate implicitly to you book or your notes.
Because applications are difficult for many of you, you want to make sure you understand every step in the Examples in Section 3.7.
Now you can go to the Session 11 assignments to continue your assignment.