\begin{tabular}{l} I ind the derivative of the function by first expanding the oxpression, \\ \( \qquad f^{\prime}(x)=(6 x-9)^{2} \) \\ \hline\( \square \)\end{tabular}

To find the derivative of the function \( f(x) = (6x - 9)^2 \) by first expanding the expression, we will follow these steps: 1. **Expand the expression**: We will use the formula for squaring a binomial, which is \( (a - b)^2 = a^2 - 2ab + b^2 \). 2. **Differentiate the expanded expression**: Once we have the expanded form, we will differentiate it with respect to \( x \). Let's start with the expansion of \( f(x) = (6x - 9)^2 \). ### Step 1: Expand the expression Using the binomial expansion: \[ f(x) = (6x - 9)^2 = (6x)^2 - 2(6x)(9) + (9)^2 \] Calculating each term: - \( (6x)^2 = 36x^2 \) - \( -2(6x)(9) = -108x \) - \( (9)^2 = 81 \) Putting it all together: \[ f(x) = 36x^2 - 108x + 81 \] ### Step 2: Differentiate the expanded expression Now we will differentiate \( f(x) = 36x^2 - 108x + 81 \). The derivative \( f'(x) \) is calculated as follows: \[ f'(x) = \frac{d}{dx}(36x^2) - \frac{d}{dx}(108x) + \frac{d}{dx}(81) \] Calculating each derivative: - \( \frac{d}{dx}(36x^2) = 72x \) - \( \frac{d}{dx}(108x) = 108 \) - \( \frac{d}{dx}(81) = 0 \) Putting it all together: \[ f'(x) = 72x - 108 \] Thus, the derivative of the function \( f(x) = (6x - 9)^2 \) is: \[ f'(x) = 72x - 108 \]