Question

The derivative \[\frac {df(x)}{dx}\] is defined as ______.

Options

• \[\lim_{\Delta x\to\infty}\frac{f(x+\Delta x)-f(x)}{\Delta x}\]

• \[\lim_{\Delta x\to0}\frac{f(x+\Delta x)+f(x)}{\Delta x}\]

• \[\lim_{\Delta x\to0}\frac{f(x+\Delta x)-f(x)}{\Delta x}\]

• \[\lim_{\Delta x\to0}f(x)\cdot\Delta x\]

Answer 1

The derivative \[\frac {df(x)}{dx}\] is defined as \[\lim_{\Delta x\to0}\frac{f(x+\Delta x)-f(x)}{\Delta x}\].

Explanation:

This is the standard limit definition of a derivative. As Δx approaches zero, the ratio \[\frac {Δy}{Δx}\]​ gives the instantaneous rate of change of f(x) at any point x.