Differentiation rules

Section 4.3 Differentiation rules

Even though in principle we can calculate the derivative of any differentiable function from its definition as the limit of the difference quotient, in practice such limit calculations quickly become rather painful. We need to find a better way of calculating derivatives. This is what we study in this section and in the next few sections. We prove a number of rules, collectively called “differentiation rules”, which allow us to drastically simplify the calculation of derivatives of complicated functions. We prove these rules using the definition of the derivative as a limit: but once the rules are proven, we can use them directly to calculate derivatives of complicated functions.

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We start in this section by studying the following foundational differentiation rules: the power rule, the constant multiple rule, the sum and difference rules, the product rule, and the quotient rule.

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Objectives

You should be able to:

Derive foundational rules of differentiation (such as power rule, constant multiple rule, sum rule, product rule, quotient rule) from the definition of the derivative of a function.
Recall foundational rules of differentiation (such as power rule, constant multiple rule, sum rule, product rule, quotient rule), and use them to calculate derivatives.

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Subsection 4.3.1 Instructional video

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Subsection 4.3.2 Key concepts

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Concept 4.3.1. Power rule.

For any real number $a \in \mathbb{R}\text{,}$

\frac{d}{d x} (x^{a}) = a x^{a - 1} .

$\begin{equation*} \frac{d}{dx} \left( x^a \right) = a x^{a-1}. \end{equation*}$

Note that it follows from the power rule that the derivative of any constant $c$ is always zero.

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Concept 4.3.2. Constant multiple rule.

For any constant $c$ and differentiable function $f\text{,}$

$\begin{equation*} \frac{d}{dx} \left( c f(x) \right) = c f'(x). \end{equation*}$

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Concept 4.3.3. Sum and difference rules.

For any two differentiable functions $f$ and $g\text{,}$

$\begin{equation*} \frac{d}{dx} \left( f(x) \pm g(x) \right) = f'(x) \pm g'(x). \end{equation*}$

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Concept 4.3.4. Product rule.

For any two differentiable functions $f$ and $g\text{,}$

$\begin{equation*} \frac{d}{dx} \left( f(x) g(x) \right)= f'(x) g(x) + f(x) g'(x). \end{equation*}$

Note that this is not equal to the product of the derivatives $f'(x) g'(x)\text{.}$

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Concept 4.3.5. Quotient rule.

For any two differentiable functions $f$ and $g\text{,}$

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

Note that this is not equal to the quotient of the derivatives

$\begin{equation*} \frac{f'(x)}{g'(x)}. \end{equation*}$

An easy way to remember the quotient rule is to sing the song:

low d-hi minus hi d-low, draw the line and square below!

Finally, note that the quotient rule is actually a consequence of the product rule and the chain rule, as we will see shortly.

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MATH 144: Calculus for the Physical Sciences I

Section 4.3 Differentiation rules

Objectives

Subsection 4.3.1 Instructional video

Subsection 4.3.2 Key concepts

Concept 4.3.1. Power rule.

Concept 4.3.2. Constant multiple rule.

Concept 4.3.3. Sum and difference rules.

Concept 4.3.4. Product rule.

Concept 4.3.5. Quotient rule.

Further readings 4.3.3 Further readings