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Given the function $f(x) = x^{-1/3}$, find the differential of $f (df)$.
Use the result to approximate: $\dfrac{1}{^3\sqrt {7.952}}$.

I found the differential, $-\dfrac{1}{3x^{4/3}}$, but I'm not totally sure how to do the second part. (is my first part right?) Thanks,

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    It must be $\;-\frac13x^{\color{red}-4/3}\;$ – 2017-02-13
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    @amWhy Yes indeed...the result was edited, though. Originally it had either a wrong result or an unclear one (because of the typing). – 2017-03-13
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    Well that explains it, then! :-) Perhaps all these comments should go, as they seem to no longer be relevant. – 2017-03-13

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For such a function we have

$$f(x+\Delta_x)\approx f(x)+f'(x)\Delta_x$$

in your case, we can choose:

$$\begin{cases}x=8\\{}\\\Delta_x=-0.048\end{cases}\;\;\implies f(7.952)\approx f(8)+f'(8)\left(-0.048\right)=8^{-1/3}-\frac138^{-4/3}\left(-0.048\right)=$$

$$=\frac12-\frac13\frac1{16}(-0.048)=\frac12+\frac13\frac3{1000}=0.501$$

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    You won by a couple of seconds ! Cheers. – 2017-02-13
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    @ClaudeLeibovici Merci, mom ami. Salut! – 2017-02-13
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    Nice answer, +1. Just want to check you are happy to use both the 'full stop; and the 'comma' as use in the decimal point mark. I know both are used, but are both used collectively in an answer? – 2017-02-13
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    @Bacon If you mean the use of period or comma to separate integer and decimal parts: both are used in different parts of the world. I use period "." but now I see that I have a typo there and I'll change. Thanks. – 2017-02-13