Find \(h^{\prime}(x)\) using the rules for differentiation: \[ h(x)=2 x^{2}+4 x-1 \] [solved]

Question

Find \(h^{\prime}(x)\) using the rules for differentiation:
\[
h(x)=2 x^{2}+4 x-1
\]

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:

\[
\begin{aligned}
\frac{d}{d x}\left(2 x^{2}\right) &=2(2) x^{(2-1)} \\
&=4 x^{1}
\end{aligned}
\]
We repeat this process for each of the remaining terms that make up the given function.
Therefore, the final answer is:
\[
h^{\prime}(x)=4 x+4
\]
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:
\[
\begin{aligned}
\frac{d}{d x}[-3] &=\frac{d}{d x}\left[-3 \cdot x^{0}\right] \\
&=0 \cdot\left[-3 \cdot x^{0-1}\right] \quad \text { multiply by the } \\
&=0 \cdot\left[\frac{-3}{x}\right] \\
&=0
\end{aligned}
\]