Implicit Differentiation, Part 2

Return to Course Outline

Previous Lecture

Anything You Want to Talk About?

Does the reference to “differentiation rules” in problem set 4 question 2 include the quotient and product rules? Yes, you can use those rules. There’s one question where I think you pretty much have to use one of them, and another where you have a choice to do so or not.

Implicit Differentiation

From section 3.8 in the textbook and the implicit differentiation discussion.

Implicit Differentiation with Other Rules

For example, find dy/dx given the equation e^xy = 1.

Start, as in all implicit differentiations, by differentiating both sides of the equation:

Then rearrange terms to isolate dy/dx:

Application: Tangent Lines

The equation x² / 4 + y² / 9 = 1 graphs as an ellipse extending ± 2 units left or right of the origin and ± 3 unit above or below it:

What is the slope of the tangent (actually, tangents) to this ellipse at x = 1?

The slope of the tangent is the derivative dy/dx. Use implicit differentiation to find that derivative:

Once we found the derivative, we had to evaluate it at x = 1. This requires plugging in values for both x and y. The value for x is easy, but to get the values for y we had to plug x = 1 into the original ellipse equation and solve for y:

When we finally found the y values corresponding to x = 1, we could plug them into the expression for the derivative:

The two values we got correspond to the fact that there are two tangents to the ellipse at x = 1: one with a negative slope in the upper right quadrant, and one with positive slope in the lower right.

Next

Derivatives of inverse functions.

We’ll do this via implicit differentiation, which is a little different from how the textbook does it.

So there’s no new reading.

But there is a discussion in which to try out some of the ideas by class time Friday.

Next Lecture