Tools for computing derivatives

Unit 4.1 Tools for computing derivatives

There are two kinds of tools for computing derivatives. We’ll use the following terminology to keep them straight.

formulas 🔗: Formulas are statements about the derivative of a particular function. For example, there is a formula for the derivative of the sine function: \(\frac{d}{dx}\sin(x)=\cos(x)\text{.}\)
rules 🔗: Rules are ways to combine information about derivatives of some functions in order to compute the derivatives of other functions.

The idea will be to leverage the formulas to help us compute the derivative of any function which can be built from our stable of basic functions.

🔗

The rules.

If we build a function out of others by adding, subtracting, multiplying, or composing, and we happen to know the derivatives of all the parts, we can express the derivative of the function we’ve built in terms of the derivatives of the parts.

🔗

Proposition 4.1. sum rule.

Let \(f\) and \(g\) be differentiable functions. Then \((f+g)' = f' + g'.\)

🔗

Proposition 4.2. difference rule.

Let \(f\) and \(g\) be differentiable functions. Then \((f-g)' = f' - g'.\)

🔗

Proposition 4.3. multiplication by a constant.

Let \(f\) be a differentiable function and \(c\) be a constant. Then \((cf)' = c f'\text{.}\)

🔗

Checkpoint 77.

Using the three propositions above, as well as formulas you’ve worked out earlier, compute the derivative of \(x - 3 \sqrt{x}\text{.}\)

🔗

Answer.

\(1-\frac{3}{2\sqrt{x}}\)

🔗

Proposition 4.4. product rule.

Let \(f\) and \(g\) be differentiable functions. Then \((fg)' = f' g + g' f\text{.}\)

🔗

Proposition 4.5. quotient rule.

Let \(f\) and \(g\) be differentiable functions. Then for any \(x\) such that \(g(x) \neq 0\text{,}\)

\begin{equation*} \frac{d}{dx} \frac{f(x)}{g(x)} = \frac{g f' - f g'}{g^2} \, \end{equation*}

all functions on the right-hand side evaluated at \(x\text{.}\)

🔗

Proposition 4.6. chain rule.

Let \(f\) and \(g\) be differentiable functions. Let \(a\) be a real number inside an open interval in the domain of \(g\) such that \(g(a)\) is inside an open interval in the domain of \(f\text{.}\) Then

\begin{equation*} \left. \frac{d}{dx} f(g(x)) \right |_{x=a} = \left ( \left. \frac{df}{dx} \right |_{x=g(a)} \right ) %%% \cdot \left ( \left. \frac{dg}{dx} \right |_{x=a} \right ) \, \end{equation*}

We can write this more compactly in prime notation as

\begin{equation*} (f \circ g)' (x) = f'(g(x)) g'(x)\ . \end{equation*}

🔗

The formulas.

We list a few formulas that are either obvious from the definition or are ones you’ve worked out already.

🔗

Proposition 4.7. some easy derivatives.

Let \(c\) be any real constant. Then,

\begin{equation*} \frac{d}{dx} \; c = 0 \end{equation*}

\begin{equation*} \frac{d}{dx} \; cx = c \end{equation*}

\begin{equation*} \frac{d}{dx} \; x^2 = 2x \end{equation*}

\begin{equation*} \frac{d}{dx} \; \sqrt{x} = \frac{1}{2 \sqrt{x}} \mbox{ for } x > 0 \, . \end{equation*}

🔗

Checkpoint 78.

Which functions \(f\) have the property that \(f'\) is a constant function? Sketch the graph of \(f\) in the case that \(f'\) is the constant function \(1/2\text{.}\)

🔗

Proposition 4.8. powers and transcendental functions.

In the following list, if no restrictions are given on \(x\text{,}\) then the statement holds for all real \(x\text{.}\)

\begin{equation*} \frac{d}{dx} \; x^n = n x^{n-1} \mbox{ when } n \mbox{ is a positive integer} \end{equation*}

🔗
\begin{equation*} \frac{d}{dx} \; x^r = r x^{r-1} \mbox{ when } x \neq 0 \mbox{ and } r \mbox{ is any nonzero real number} \end{equation*}

🔗
\begin{equation*} \frac{d}{dx} \; e^x = e^x \end{equation*}

🔗
\begin{equation*} \frac{d}{dx} \; a^x = a^x \cdot \ln a \mbox{ for } a > 0 \mbox{ and all real } x \end{equation*}

🔗
\begin{equation*} \frac{d}{dx} \; \ln x = \frac{1}{x} \mbox{ for } x > 0 \end{equation*}

🔗
\begin{equation*} \frac{d}{dx} \; \sin x = \cos x \end{equation*}

🔗
\begin{equation*} \frac{d}{dx} \; \cos x = - \sin x \end{equation*}

🔗
\begin{equation*} \frac{d}{dx} \; \tan x = \sec^2 x \mbox{ when this is finite} \end{equation*}

🔗
\begin{equation*} \frac{d}{dx} \; \arcsin x = \frac{1}{\sqrt{1-x^2}} \end{equation*}

🔗
\begin{equation*} \frac{d}{dx} \; \arccos x = \frac{-1}{\sqrt{1-x^2}} \end{equation*}

🔗
\begin{equation*} \frac{d}{dx} \; \arctan x = \frac{1}{1+x^2} \end{equation*}

🔗

🔗

Checkpoint 79.

Compute the slope of the function \(f(x) := a^x\) at \(x=0\text{.}\)

🔗

For which \(a\) is this slope equal to 1?

🔗

Is this consistent with the formula given for the derivative of \(e^x\) given in Proposition 4.8?

🔗

Answer 1.

\(\ln\mathopen{}\left(a\right)\)

🔗

Answer 2.

\(e\)

🔗

Checkpoint 80.

Let \(f(x) := x^{-1}\) and \(g(x) := x^3\text{.}\) This exercise takes you step by step through a test of the product rule.

What is \(f'\text{?}\)
🔗

🔗
What is \(g'\text{?}\)
🔗

🔗
what is \((f')(g')\text{?}\)
🔗

🔗
What is \(fg\text{?}\)
🔗

🔗
What does the product rule give you for \((fg)'\text{?}\)
🔗

🔗
What do you get for \((fg)'\) by first multiplying, then using rule 1 from Proposition 4.8(the power rule)?
🔗

🔗

🔗

Answer 1.

\(-x^{-2}\)

🔗

Answer 2.

\(3x^{2}\)

🔗

Answer 3.

\(-3\)

🔗

Answer 4.

\(x^{2}\)

🔗

Answer 5.

\(2x\)

🔗

Answer 6.

\(2x\)

🔗

You are probably pretty experienced at taking apart algebraic expressions into sums and differences of products and quotients of simpler expressions. Here are some more exercises to check that you can do this and then apply the differentiation rules above.

🔗

Checkpoint 81.

Use the sum, difference, product and quotient rules, along with derivative formulas given in Proposition 4.8, to evaluate \(f'(x)\) in each of these cases.

\(f(x) := x^3 e^x\)
🔗

🔗
\(f(x) := \frac{1}{x^{2.5}}\)
🔗

🔗
\(f(x) := x \ln x - x\)
🔗

🔗
\(f(x) := x \arcsin x\)
🔗

🔗

🔗

Answer 1.

\(\left(3x^{2}+x^{3}\right)e^{x}\)

🔗

Answer 2.

\(-2.5x^{-3.5}\)

🔗

Answer 3.

\(\ln\mathopen{}\left(x\right)\)

🔗

Answer 4.

\(\sin^{-1}\mathopen{}\left(x\right)+\frac{x}{\sqrt{1-x^{2}}}\)

🔗

Taking apart algebraic expressions into compositions of functions, as is needed for the chain rule, can be a little trickier.

🔗

Example 4.9.

In order to differentiate \((1+x^2)^{1/3}\) you need to recognize this as a composition \(f(g(x))\) with \(f(x) = x^{1/3}\) and \(g(x) = 1 + x^2\text{.}\) The chain rule tells us that the derivative of \((1+x^2)^{1/3}\) at \(x=a\) will be given by

\begin{equation} \left ( \left. \frac{d}{dx} x^{1/3} \right |_{x=1+a^2} \right ) \; \cdot \; \left ( \left. \frac{d}{dx} (1+x^2) \right |_{x=a} \right ) \, .\tag{4.1} \end{equation}

The derivative of \(x^{1/3}\) is \((1/3) x^{-2/3}\) by the power rule (the second identity in Proposition 4.8); the derivative of \(1+x^2\) is \(0 + 2x = 2x\) by the sum rule and the power rule. This shows (4.1) to equal

\begin{equation*} \left ( \left. \frac{1}{3} x^{-2/3} \right |_{x=1+a^2} \right ) \cdot \left ( \left. 2x \right |_{x=a} \right ) \\ = \frac{1}{3} (1+a^2)^{-2/3} (2a) \, \end{equation*}

🔗

The next few exercises check on your understanding of the chain rule. The first two tell you how to choose \(f\) and \(g\text{.}\) The last two do not.

🔗