The derivative of a function

Section 4.1 The derivative of a function

In our overview of calculus, we realized that the derivative of a function was a fundamental concept: given the position function of an object, it ouputs its velocity function. Generally, it gives the instantaneous rate of change of a function. Geometrically, the derivative of a function at a point gives the slope of the tangent line to the graph of the function at that point.

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We have also seen that the derivative of a function is defined as a limit. Now that we know how to evaluate limits, we can go back to the concept of derivatives, and study it in more depth!

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Objectives

You should be able to:

Explain why the derivative of a function is itself a function.
Calculate the derivative of simple functions (such as linear and quadratic polynomials, square root function, and simple rational functions) from the definition of the derivative of a function.
Sketch the graph of the derivative of a function from the graph of the function itself.

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Subsection 4.1.1 Instructional video

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Subsection 4.1.2 Key concepts

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Concept 4.1.1. The derivative of a function.

The derivative of a function $f(x)$ is defined by

f^{'} (x) = lim_{h \to 0} \frac{f (x + h) - f (x)}{h} .

$\begin{equation*} f'(x) = \lim_{h \to 0} \frac{f(x+h) - f(x)}{h}. \end{equation*}$

We can calculate the derivative of any function directly from the definition, by evaluating the limit above.

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Concept 4.1.2. Notation.

The following are equivalent notations for the derivative of a function $y=f(x)\text{:}$

f^{'} (x) = y^{'} = \frac{d y}{d x} = \frac{d f}{d x} = \frac{d}{d x} f (x) .

$\begin{equation*} f'(x) = y' = \frac{dy}{dx} = \frac{df}{dx} = \frac{d}{dx} f(x). \end{equation*}$

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Concept 4.1.3. Interpretation of the derivative.

The value of $f'(x)$ of $x=a$ represents:

The instantaneous rate of change of $f(x)$ at $x=a$ (instantaneous velocity if $f$ is a position function);
The slope of the tangent line to the graph of $y=f(x)$ at $(a, f(a))\text{.}$

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Concept 4.1.4. Higher derivatives.

The second derivative of $f$ is denoted by $f''(x)$ or
$\begin{equation*} \frac{d^2 f}{dx^2}. \end{equation*}$
It is the derivative of $f'\text{,}$ that is, the derivative of the derivative of $f\text{.}$ For instance, if $f$ is a position function, $f'$ is the velocity function and $f''$ is the acceleration function.
Similarly, the third derivative of $f$ is denoted by $f'''(x)$ or
$\begin{equation*} \frac{d^3 f}{dx^3}. \end{equation*}$
Is it the derivative of $f''\text{.}$
In general, the $n$ 'th derivative of $f$ is denoted by $f^{(n)}(x)$ or
$\begin{equation*} \frac{d^n f}{dx^n}. \end{equation*}$
It is the derivative of $f^{(n-1)}\text{,}$ that is, it is obtained from $f$ by differentiating $n$ times.

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MATH 144: Calculus for the Physical Sciences I