Tutorial: The chain rule

Game version

This tutorial: Part A: Chain rule: Basics

Go to Part B: Chain rule and Leibniz notation

(This topic is also in Section 4.4 in Applied Calculus or Section 11.4 in Finite Mathematics and Applied Calculus)

#[I don't like this new tutorial. Take me back to the old tutorial!][No me gusta este nueve tutorial. ¡Regresame al tutorial más viejo!]#

Adaptive game tutorial
This adaptive game tutorial will help you master this topic in a way that adapts to your ability as you practice the questions.
If this is your first time studying the material, you may find yourself requiring more help, and as a result your game scores could end up lower, or you could even "die." Do not be discouraged! Just press "New game" to play the game as many times as you need in order to require less help and improve your scores.
You can also try the non-game version (link on the side), which does not score you, is not randomized or adaptive and gives you all the answers, but will not give you additional practice at the level you might need.
To complete this game you must answer all the questions correctly; you will be allowed to have a number of tries for some questions.
The questions are randomized: Expect to see lots of differences differences every time you load the page.
Press "Scores" at any time to show the scores you currently have.
The game is automatically saved. Relaunching the game on the same computer with the same browser will bring up the saved version.
Press "New game" to discard the saved game and start a new game.
Caution: Clicking on the little pictures that appear from time to time on the left can have unexpected consequences! You might want to experiment with them; this is a game after all!

Warmup: The calculation throught experiment

#[Let us start with a quick review and quiz on using the "Calculation Thought Experiment (CTE)" discussed in the %%prevsectut:][Empecemos con un repaso y concurso rápido sobre el uso del "Experimento mental de cálculo (EMC)" descrito en el %%prevsectut:]#

Calculation thought experiment (CTE)

The calculation thought experiment is a technique to determine whether to treat an algebraic expression as a product, quotient, sum, difference, or power:

last

* #[See the %%orderoperations for a discussion of order of operations][Ve el %%orderoperations para una discusión del orden de operactiones]#.

%%Examples

1. $(3x^2-4)(2x+1)$ is calculated by first calculating the expressions in parentheses and then multiplying. Since the last step is multiplication, we treat the expression as a product. Thus, when calculating its derviative, we use the product rule.

2. $-7(2x+1)$ is calculated by first calculating the expression in parentheses and then multiplying by $-7$, so it too is a product, and in particular a constant multiple as we are mutliplying by the constant $-7$. We can therefore calculate its derivative using the constant multiple rule more easily than using the product rule.

3. $\dfrac{2x-1}{x}$ is calculated by first calculating the numerator and denominator separately, and then dividing one by the other. Since the last step is division, we treat the expression as a quotient, and so, in calculating its derivative, we use the product rule.

4. $(4x-1)(x+2) + x^2$ is calculated by first calculating the product $(4x-1)(x+2)$, then calculating $x^2$, and finally adding the two answers. Since the last step is addition, we treat the expression as a sum, and thus we use the rule for sums when taking its derivative.

5. $(3x^2-1)^5$ is calculated by first calculating the expression in parentheses, and then raising the answer to the fifth power. Since the last step is raising to a power, we treat the expression as a power. Below we see how to take derivatives of powers of functions other than $x$.

Some for you

Introducing the chain rule

%%Q So what, exactly, is the chain rule all about?
%%A Here is an example: We know that the derivative of $x^3$ is $3x^2$. What then would you say is the derivative of something more complicated raised to the third power, like for instance $(2x + 3x^4)^3$?

%%Q Duh! That's just $3(2x + 3x^4)^2$.
%%A No it isn't; the power rule applies only to powers of $x$, and not powers of more complicated things. In this example, you could find the derivative by first expanding

$(2x+3x^4)^3 = (2x+3x^4)(2x+3x^4)(2x+3x^4) = 8x^3+36x^6+54x^9+27x^12$

and then taking the derivative using the power rule on each term.

%%Q OK then; what about $(2x + 3x^4)^{100}$? Are we really expected to expand that just to find its derivative?
%%A Luckily for us, no; and it is the chain rule that tells us how to quickly and easily calculate the derivatives of expressions like that.

In what follows, think of, say, $(2x + 3x^4)^{100}$ as $u^{100}$, where $u$ is the "inside:" $u = 2x+3x^4$.

Chain rule

If $u$ is a differentiable function of $x$, and $f$ is a differentiable function of $u$, then $f$ is a differentiable function of $x$, and:

$\displaystyle \frac{d}{dx}[f(u)] = f'(u) \frac{du}{dx}$

Chain rule in words:

The derivative of $f$ of a quantity is the derivative of $f$, evaluated at that (original) quantity, times the derivative of the quantity.

%%Examples

1. \gap[20] \t $\displaystyle \frac{d}{dx}[u^3] = 3u^2 \frac{du}{dx}$ \gap[30] #[Chain rule with][Relga de la cadena con]# $f(x) = x^3$

The derivative of a quantity cubed is 3 times the (original) quantity squared, times the derivative of the quantity.

Eg.

\t $\displaystyle \frac{d}{dx}[\color{indianred}{(2x + 3x^4)}^3]$ \t

$\displaystyle {}= 3\color{indianred}{(2x + 3x^4)}^2 \frac{d}{dx}\color{indianred}{(2x + 3x^4)}$

$\dfrac{d}{dx}\text{(quantity)}^3 = 3\text{(quantity)}^2\dfrac{d}{dx}\text{(quantity)}$

\\ \t \t $\displaystyle {}=3(1+x^2)^2 \cdot 2x$ \\ \t \t $\displaystyle {}=6x(1+x^2)^2$

2. \gap[20] \t $\displaystyle \frac{d}{dx}[u^n] = nu^{n-1} \frac{du}{dx}$ \gap[30] Chain rule with $f(x) = x^n$

Generalized power rule: The derivative of a quanity raised to the power n is n times the (original) quantity raised to the n − 1, times the derivative of the quantity.

Eg.

\t $\displaystyle \frac{d}{dx}\left[\frac{2}{\color{indianred}{(-x+x^3)}^3}\right]$ \t

$\displaystyle {}= \frac{d}{dx}\left[2\color{indianred}{(-x+x^3)}^{-3}\right]$

#[Converted to %%powerform][Convertido en %%powerform]#

\\ \t \t

$\displaystyle {}=(2)(-3)\color{indianred}{(-x+x^3)}^{-4}\frac{d}{dx}\color{indianred}{(-x+x^3)}$

$\dfrac{d}{dx}\text{(quantity)}^{-3} = -3\text{(quantity)}^{-4}\dfrac{d}{dx}\text{(quantity)}$

\\ \t \t $\displaystyle {}=-6(-x+x^3)^{-4}(-1+3x^2)$ \\ \t \t $\displaystyle {}=\frac{-8(-1+3x^2)}{-x+x^3}$

3. \gap[20] \t $\dfrac{d}{dx}\sqrt{u} = \dfrac{1}{2\sqrt{u}}\dfrac{du}{dx}$ \gap[30] Chain rule with $f(x) = \sqrt{x}$

The derivative of the square root of a quantity is 1 over twice the square root of the (original) quantity, times the derivative of the quantity.

Eg.

\t $\displaystyle \frac{d}{dx}\sqrt{\color{indianred}{x^2-1}}$ \t

$\displaystyle {}=\frac{1}{2\sqrt{\color{indianred}{x^2-1}}}\frac{d}{dx}\color{indianred}{(x^2-1)}$

$\dfrac{d}{dx}\sqrt{\text{quantity}} = \frac{1}{2\sqrt{\text{quantity}}}\dfrac{d}{dx}(\text{quantity})$

\\ \t \t $\displaystyle {}=\frac{1}{2\sqrt{x^2-1}}\cdot (2x)$ \\ \t \t $\displaystyle {}=\frac{x}{\sqrt{x^2-1}}$

4. \gap[20] \t $\dfrac{d}{dx}|u| = \dfrac{|u|}{u}\dfrac{du}{dx}$ \gap[30] Chain rule with $f(x) = |x|$

The derivative of the absolute value of a quantity is the absolute value of the (original) quantity over the quantity, times the derivative of the quantity.

Eg.

\t $\displaystyle \frac{d}{dx}|\color{indianred}{4x-5}|$ \t

$\displaystyle {}=\frac{|\color{indianred}{4x-5}|}{\color{indianred}{4x-5}}\frac{d}{dx}\color{indianred}{(4x-5)}$

$\dfrac{d}{dx}|\text{quantity}| = \dfrac{|\text{quantity}|}{\text{quantity}}\dfrac{d}{dx}(\text{quantity})$

\\ \t \t $\displaystyle {}=\frac{|4x-5|}{4x-5}\cdot (4)$ \\ \t \t $\displaystyle {}=\frac{4|4x-5|}{4x-5}$

Combining the rules for differentiation

%%Q How do we deal with more complicated expressions that are combinations of products or quotients but also appear to require the chain rule, like

$\displaystyle y = \left(\frac{x^2}{x-1} + \sqrt{3x^2-1}\right)^4$? %%A It is in cases like this that we go back to our old friend the CTE (Calculation Thought Experiment), which tells us which rule to use at each step. Following is a quiz on using the CTE for this purpose, just as we did in the %%prevsectut.

Now that you are an expert on taking derivatives of all sorts of things, here is a final quiz to challenge you.

Now try the exercises in Section 4.4 in Applied Calculus or Section 11.4 in Finite Mathematics and Applied Calculus.