Overview
In several different contexts, we will see that it is useful to consider a function that is described implicitly, rather than explicitly. A prominent simple example is that of a circle: while no single function of the form \(y = f(x)\) can represent every point on the circle, there is nonetheless a key relationship between the \(x\) and \(y\) coordinates of points on the curve. In more complicated settings, we’ll see that while there is a relationship between \(x\) and \(y\), there is no way to explicitly solve for \(y\) in terms of \(x\). In this setting and others similar to it, we want to be able to still compute \(\frac{dy}{dx}\). The process of implicit differentiation enables us to do so.
Basic learning objectives
These are the tasks you should be able to perform with reasonable fluency when you arrive at your next class meeting. Important new vocabulary words are indicated in italics.
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(Review) Given a basic differentiation rule, give its chain rule version. For example, the chain rule version of \(\frac{d}{dx} [x^n] = nx^{n-1}\) is:
$$\frac{d}{dx} [y^n] = ny^{n-1}\cdot \frac{dy}{dx} \qquad \text{or} \qquad \frac{d}{dx} [y^n] = ny^{n-1}\cdot y'$$ -
Give an example of a function that is defined explicitly by a relation between \(x\) and \(y\), and give an example where the relation is implicit.
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Given an equation in \(x\) and \(y\), where \(y\) is an implicit function of \(x\), determine if a given point lies on the graph of the equation.
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Be comfortable implicitly differentiating basic expressions such as \(\frac{d}{dx}[x^2 f(x)]\), where \(f\) does not have a known formula.
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Understand the difference between the notations “\(\frac{d}{dx}[\ \ ]\)” and “\(\frac{dy}{dx}\).”
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Use the different notations \(\frac{dy}{dx}\) and \(\frac{dy}{dx}|_{(a,b)}\) appropriately, given a certain context.
Advanced learning objectives
In addition to mastering the basic objectives, here are the tasks you should be able to perform after class, with practice:
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Determine \(\frac{dy}{dx}\) via implicit differentiation for complicated curves such as \(x^3 + 6xy + y^3 = 1\) and \(\sin(y) + y = x^3 + x\).
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Find the slope and equation of a tangent line to a curve that is specified by an equation that is not the graph of a function.
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Given an implicit curve and a point on the curve, find the local linearisation at this point to approximate the coordinates of nearby points.
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Understand how to use \(\frac{dy}{dx}\) (which may depend on both \(x\) and \(y\)) to determine all points where the tangent line to a given implicit curve is horizontal or vertical.
To prepare for class
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Read the beginning of section 2.7 (stop just before Activity 2.7.2).
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Watch the following video:
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Do the Preview Activity for section 2.7 (on WeBWorK if required by your teacher).
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Watch the following videos (the second was created at CCSL). Note that the example at the end of the first video is the same as the one referred to in the second video at around 4:09, and can be found in example 2.7.3 in Active Calculus):
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Write a short explanation (and include this in your feedback post) of the difference between writing \(\frac{d}{dx}[x^2 + y^2]\) and \(\frac{dy}{dx}[x^2 + y^2]\). That is, explain the big difference between the meanings of the symbols \(\frac{d}{dx}\) and \(\frac{dy}{dx}\).
After class
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Watch the following videos for some more calculation examples (the first video was created at CCSL):
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Do some experimentation with the following interactive applet: Implicit Differentiation
