Derivatives of Inverse Functions, Part 2

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The next session is tomorrow from 4:00 - 5:30. Watch for an announcement with Zoom info around 3:00.

Problem Set 6

There’s a new problem set, mostly on derivatives of exponential functions, inverse functions, and implicit differentiation.

Work on it this week, and grade it late in the week or early next week.

Wrapping Up Inverse Functions

Building on the discussion of derivatives of inverse functions.

Roots

What’s the derivative of the nth root of x? More generally, try to show that the power rule applies to powers of the form 1/n for any positive integer n.

This is easier to work out using implicit differentiation than using the general formula for derivatives of inverse functions. To start, rewrite the equation for y and x in terms of a function we know how to differentiate:

Then use implicit differentiation to find dy/dx:

Then finally simplify and rewrite the derivative in terms of x. To do this, we use the original equation y = x^1/n to substitute x^1/n for y, and an algebra rule about powers of powers:

The final form we got for the derivative is exactly what the power rule would have given if applied to x^1/n, so now we know that the power rule works for powers that are reciprocals of positive integers as well as for positive and negative integer powers.

Antiderivatives

Some of the above derivative formulas also correspond to important antiderivative formulas.

For example...

Mathematica and Implicit Equations

Only indirectly related to inverse functions, but here’s a useful thing to know about Mathematica now that you know about implicit definitions of functions.

Mathematica’s Plot function is good for plotting functions defined by an explicit equation, but how do you plot the points that satisfy an implicit equation?

The answer is the ContourPlot function. A generic use looks like

ContourPlot[ equation, {x,lowX,highX}, {y,lowY,highY}]

where equation is the equation that relates x and y, lowX is the low bound on the x values you want to plot over and highX is the high bound, and lowY and highY are the bounds for the range of y coordinates.

For example, to plot the hyperbola x² - y² = 1 for -2 ≤ x ≤ 2 and -2 ≤ y ≤ 2, you would say

ContourPlot[ x^2 - y^2 == 1, {x, -2, 2}, {y, -2, 2}]

Be careful that equality in an equation has to be written as two equal signs, i.e., ==!

Here is a Mathematica notebook that demonstrates the ContourPlot function and contrasts it to Plot.

Next

You now know pretty much all the differentiation rules you’ll need. So we can start looking at some applications of derivatives.

The first is in solving so-called “related rates” problems.

Please read the introduction to section 4.1 of the textbook, and “Setting Up Related-Rates Problems” in section 4.1 by class time Wednesday.

Please also participate in this discussion of related rates problems by class time Wednesday.

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