Questions: Find (f^prime(x)). [f(x)=frac4 x-96 x+7] [f^prime(x)=square]

$Find (f^prime(x)). [f(x)=frac4 x-96 x+7] [f^prime(x)=square]$

Transcript text: Find $f^{\prime}(x)$. \[ \begin{array}{l} f(x)=\frac{4 x-9}{6 x+7} \\ f^{\prime}(x)=\square \end{array} \]

Solution

Solution Steps

To find the derivative $ f^{\prime}(x) $ of the function $ f(x) = \frac{4x - 9}{6x + 7} $, we can use the quotient rule. The quotient rule states that if you have a function $ f(x) = \frac{u(x)}{v(x)} $, then its derivative is given by:

\[ f^{\prime}(x) = \frac{u^{\prime}(x)v(x) - u(x)v^{\prime}(x)}{(v(x))^2} \]

where $ u(x) = 4x - 9 $ and $ v(x) = 6x + 7 $. We need to find the derivatives $ u^{\prime}(x) $ and $ v^{\prime}(x) $, and then apply the quotient rule formula.

Step 1: Define the Function

We start with the function defined as: \[ f(x) = \frac{4x - 9}{6x + 7} \]

Step 2: Apply the Quotient Rule

To find the derivative $ f^{\prime}(x) $, we apply the quotient rule: \[ f^{\prime}(x) = \frac{u^{\prime}(x)v(x) - u(x)v^{\prime}(x)}{(v(x))^2} \] where $ u(x) = 4x - 9 $ and $ v(x) = 6x + 7 $.

Step 3: Calculate Derivatives

We compute the derivatives: \[ u^{\prime}(x) = 4 \] \[ v^{\prime}(x) = 6 \]

Step 4: Substitute into the Quotient Rule

Substituting $ u^{\prime}(x) $, $ v(x) $, $ u(x) $, and $ v^{\prime}(x) $ into the quotient rule gives: \[ f^{\prime}(x) = \frac{4(6x + 7) - (4x - 9)(6)}{(6x + 7)^2} \]

Step 5: Simplify the Expression

Calculating the numerator: \[ 4(6x + 7) = 24x + 28 \] \[ (4x - 9)(6) = 24x - 54 \] Thus, the numerator becomes: \[ 24x + 28 - (24x - 54) = 24x + 28 - 24x + 54 = 82 \] So, we have: \[ f^{\prime}(x) = \frac{82}{(6x + 7)^2} \]