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Find $\frac{d}{d x} f(x)$ using the rules for differentiation:
$f(x)=-3 x^{2}-6 x+6$
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The rules for differentiation:
$\frac{d}{d x} a x^{n}=(\text { an }) x^{n-1}$
In words: for each term we multiply the coefficient by the exponent and subtract one from the exponent.
Let's consider the first term of the function and apply the rule:
\begin{aligned} \frac{d}{d x}\left(-3 x^{2}\right) &=2(-3) x^{(2-1)} \\ &=-6 x^{1} \end{aligned}
We repeat this process for each of the remaining terms that make up the given function.
$\frac{d}{d x} f(x)=-6 x-6$
Important: the derivative of a constant term, such as 3 , will always be zero.

We can apply the rule to see why this is true:
\begin{aligned} \frac{d}{d x} &=\frac{d}{d x}\left[3 . x^{0}\right] \\ &=0 .\left[3 . x^{0-1}\right] \\ &=0 .\left[\frac{3}{x}\right] \\ &=0 \end{aligned}
by Platinum (119,184 points)

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