The derivative as a function

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The derivative of a function, as a function

We know that to find the derivative of a function at a point $x=a$ we write $f'(a) = \lim _{h\to 0}\frac {f(a+h)-f(a)}{h}.$ However, if we replace the given number $a$ with a variable $x$ , we now have $f'(x) = \lim _{h\to 0}\frac {f(x+h)-f(x)}{h}.$ This tells us the instantaneous rate of change at any given point $x$ .

Given a function $f$ from the real numbers to the real numbers, the derivative $f'$ is also a function from the real numbers to the real numbers. Understanding the relationship between the functions $f$ and $f'$ helps us understand any situation (real or imagined) involving changing values.

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Phi	$\Phi$
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Omega	$\Omega$

The derivative of a function, as a function

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