woowheedr

2022-07-02

How to find instantaneous rate of change for $f\left(x\right)=\mathrm{ln}\left(x\right)$ when x=0?

Keely Fernandez

Expert

The instantaneous rate of change is also defined as the derivative, so all we're asked to do is to take a derivative using some essentials rules, such as the the following:
$\frac{d}{dx}\left[\mathrm{ln}u\right]=\frac{{u}^{\prime }}{u}$
Letting $u=x\to du=dx$, and so
${f}^{\prime }\left(x\right)=\frac{1}{x}\to {f}^{\prime }\left(0\right)=\frac{1}{0}=0$
The instantaneous rate of change is, in this case, 0.

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