Question

Find \(\frac{d}{d x} f(x)\) using the rules for differentiation:
\[
f(x)=-3 x^{2}-6 x+6
\]

Answer 1

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.
Therefore, the final answer is:
\[
\frac{d}{d x} f(x)=-6 x-6
\]
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:
\[
\begin{aligned}
\frac{d}{d x}[3] &=\frac{d}{d x}\left[3 . x^{0}\right] \\
&=0 .\left[3 . x^{0-1}\right] \\
&=0 .\left[\frac{3}{x}\right] \\
&=0
\end{aligned}
\]