6.1 Differentiability, the Tangent Line-Linear Approximation

]> 6.1 Differentiability, the Tangent Line-Linear Approximation

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6.1 Differentiability, the Tangent Line-Linear Approximation

A function $f$ of one real variable is said to be differentiable at argument $x$ , if its graph looks like a straight line for arguments in any open interval including $x$ . (An open interval is one that does not contain its endpoints.)

Its derivative at $x$ is the slope of that line.

(To be more precise, for whatever positive criterion of nearness you choose however small, there is an open interval containing $x$ so that for every $x'$ in the interval other than $x$ itself, the difference between $\frac{f (x') - f (x)}{x' - x}$ and the slope of that line is less than that criterion.) Note

The line that $f$ resembles near argument $x$ is called the tangent line to $f$ at argument $x$ and the linear function it represents is called the linear approximation to $f$ at argument $x$ .

The slope of the tangent line at $x$ is given by $\frac{d f}{d x}$ on that line for any two point $P_{2}$ and $P_{1}$ on it, with $P_{1} = (P_{1 x}, P_{1 y})$ and $P_{2} = (P_{2 x}, P_{2 y})$

d f = P_{2 y} - P_{1 y}

d x = P_{2 x} - P_{1 x}

We use the notation $d x$ and $d f$ to denote changes in the corresponding variables that are so small that we can assume the linear approximation to $f$ (and to any other function involved in the definition of $f$ ) is exacty satisfied (and if there is no such distance create one in your imagination).

Changes of this sort are called differentials . The derivative of $f$ at argument $x$ is usually written as $\frac{d f}{d x}$ or $\frac{d f (x)}{d x}$ , or $f' (x)$ .

In fact if $d f$ and $d x$ are differentials, then the derivative is $\frac{d f}{d x}$ by definition since the derivative is the ratio of the change of $f$ to the change of $x$ in the linear approximation.

The applet here allows you to enter any standard function and domain, and look at it, its slope and derivative in it.