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

To find the derivative of the function \( f(x)=6 x^{3}+7 x^{2}-9 x+4 \), we apply the power rule, which states that the derivative of \( x^n \) is \( n x^{n-1} \). Calculating each term: - The derivative of \( 6 x^{3} \) is \( 18 x^{2} \). - The derivative of \( 7 x^{2} \) is \( 14 x \). - The derivative of \( -9 x \) is \( -9 \). - The derivative of a constant (4) is 0. Putting it all together, we have: \[ f'(x) = 18 x^{2} + 14 x - 9 \] Therefore, the correct answer is \( D) 18 x^{2}+14 x-9 \). To double-check your calculations, remember that the power rule is your best friend! Mistakes often happen with the coefficients or forgetting to apply the rule correctly to each term. Ensuring clarity in each step helps in avoiding those pesky common mistakes.