Chapter Two: Section Five Implicit Differentiation

Chapter Two: Section Five Up to this point all of our derivative rules have considered functions explicitly defined in terms of x in their definitions. What do we do if our function cannot be written explicitly in the form of y = f (x) ? This is a thorny question…

Chapter Two: Section Five We need to think carefully about what the symbols we have been using actually mean in these situations. We have alternately been using two different symbols for the left hand side of the equation when we take the derivative of an equation of the form y = f (x). Let’s look at one example with two different notational choices:

Chapter Two: Section Five If y = x 3 – 4sin x + 2 is your function and you are asked to find the derivative of this function, you can write one of the following two equations:  y ‘ = 3x 2 – 4cosx  dy/dx = 3x 2 – 4cosx In each of these cases we are saying: The derivative of y is three times x squared minus 4 times the cosine of x.

Chapter Two: Section Five The key idea here is that we ARE saying “the derivative of y is…” The symbols y ‘ and dy/dx are each read as the derivative of y. Extending this idea, and using our product rule, we have to think clearly about taking the derivative of a function that is not explicitly defined for y. Let’s look at an example…

Chapter Two: Section Five Find the derivative of 2 xy + xy 2 = 6 at the point ( 2, 1 ). Now, look at the point (2,1) and we can find the numerical value of the derivative as follows:

Chapter Two: Section Five That is a pretty hard example to follow for this new idea. I understand that. Look it over carefully and be prepared to ask specific questions about the solution I provided. In class, I will provide an example that we can check using other techniques so that you can feel more confident that this idea makes sense. You can also look at the following link for some help: XI XI