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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 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 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 d f d x or d f ( x ) d x , or f ' ( x ) .

In fact if d f and d x are differentials, then the derivative is 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.

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