Given the function $g(x)=4 x^{3}+6 x^{2}-24 x$, find the first derivative, $g^{\prime}(x)$.
\[
g^{\prime}(x)=
\]

Answer

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Answer

$\boxed{g'(x) = 12x^2 + 12x - 24}$

Steps

Step 1 ：The derivative of a function can be found by applying the power rule, which states that the derivative of $x^n$ is $n \cdot x^{n-1}$.

Step 2 ：Applying the power rule to the function $g(x) = 4x^3 + 6x^2 - 24x$, we get $g'(x) = 3 \cdot 4x^{3-1} + 2 \cdot 6x^{2-1} - 24 \cdot x^{1-1}$.

Step 3 ：Simplifying the above expression, we get $g'(x) = 12x^2 + 12x - 24$.

Step 4 ：So, the first derivative of the function $g(x)$ is $g'(x) = 12x^2 + 12x - 24$.

Step 5 ：$\boxed{g'(x) = 12x^2 + 12x - 24}$