Step 1:

We notice that the given function is a quotient of two other functions. To find the derivative of this, we can use the quotient rule. First, let’s label the two functions as:

[latex]\large{h(x) = 4x^2}[/latex] and [latex]\large{g(x) = 2x-3}[/latex]

To use the quotient rule, we also need the derivative of these functions. We can easily find the derivatives as:

[latex]\large{h'(x) = 8x}[/latex] and [latex]\large{g'(x) = 2}[/latex]

Step 2:

The quotient rule formula states that:

[latex]\large{\frac{d}{dx}\frac{f(x)}{g(x)} = \frac{g(x)f'(x) – f(x)g'(x)}{g(x)^2}}[/latex]

Thus, substituting into the quotient rule formula, we have:

[latex]\large{\frac{d}{dx}(\frac{4x^2}{2x-3}) = \frac{(2x-3)(8x) \:- \:(4x^2)(2)}{(2x-3)^2}}[/latex]

Step 3:

Simplifying, we have:

[latex]\Large{ = \frac{16x^2-24x-8x^2}{(2x-3)^2}}[/latex]

[latex]\Large{ = \frac{8x^2-24x}{(2x-3)^2}}[/latex]

Thus, this is the final answer for the derivative. So:

[latex]\Large{f'(x) = \frac{8x^2-24x}{(2x-3)^2}}[/latex]

Solution Complete!