Questions: If f(x)=(7x+4)^3/(9x+2), what is f'(x)?

Transcript text: If $f(x)=\frac{(7 x+4)^{3}}{9 x+2}$, what is $f^{\prime}(x) ?$

Solution

Solution Steps

Step 1: Define the Function

We start with the function defined as: \[ f(x) = \frac{(7x + 4)^{3}}{9x + 2} \]

Step 2: Apply the Quotient Rule

To find the derivative $ f^{\prime}(x) $, we apply the quotient rule, which states: \[ f^{\prime}(x) = \frac{g(x)h^{\prime}(x) - h(x)g^{\prime}(x)}{(h(x))^{2}} \] where $ g(x) = (7x + 4)^{3} $ and $ h(x) = 9x + 2 $.

Step 3: Calculate Derivatives

We compute the derivatives:

$ g^{\prime}(x) = 21(7x + 4)^{2} $
$ h^{\prime}(x) = 9 $

Step 4: Substitute into the Quotient Rule

Substituting into the quotient rule gives: \[ f^{\prime}(x) = \frac{(9)(7x + 4)^{3} - (21(7x + 4)^{2})(9x + 2)}{(9x + 2)^{2}} \]

Step 5: Simplify the Expression

After simplification, we find: \[ f^{\prime}(x) = \frac{6(1029x^{3} + 1225x^{2} + 392x + 16)}{(9x + 2)^{2}} \]

Final Answer

Thus, the derivative of the function is: \[ \boxed{f^{\prime}(x) = \frac{6(1029x^{3} + 1225x^{2} + 392x + 16)}{(9x + 2)^{2}}} \]

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