19. Find the derivative of $$ f(x)=6 x^{3}+7 x^{2}-9 x+4 $$ $$ \begin{array}{llll} ext { A) } 36 x-14 & ext { B) } 9 x^{2}+14 x+9 & ext { C) } 36 x+14 & ext { D) } 18 x^{2}+14 x-9\end{array} $$

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19. Find the derivative of $$ f(x)=6 x^{3}+7 x^{2}-9 x+4 $$ $$ \begin{array}{llll}	ext { A) } 36 x-14 & 	ext { B) } 9 x^{2}+14 x+9 & 	ext { C) } 36 x+14 & 	ext { D) } 18 x^{2}+14 x-9\end{array} $$

UpStudy AI · Accepted Answer

To find the derivative of the function $$ f(x) = 6x^3 + 7x^2 - 9x + 4 $$, we will apply the power rule of differentiation, which states that if $$ f(x) = ax^n $$, then $$ f'(x) = n \cdot ax^{n-1} $$.

Let's differentiate each term of $$ f(x) $$:

1. The derivative of $$ 6x^3 $$ is $$ 3 \cdot 6x^{3-1} = 18x^2 $$.
2. The derivative of $$ 7x^2 $$ is $$ 2 \cdot 7x^{2-1} = 14x $$.
3. The derivative of $$ -9x $$ is $$ -9 $$.
4. The derivative of the constant $$ 4 $$ is $$ 0 $$.

Now, we can combine these results:

$$
f'(x) = 18x^2 + 14x - 9
$$

Thus, the derivative of $$ f(x) $$ is $$ 18x^2 + 14x - 9 $$.

The correct answer is:

**D) $$ 18x^2 + 14x - 9 $$**.
The derivative of $$ f(x) $$ is $$ 18x^2 + 14x - 9 $$.

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