Step 1:

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

[latex]\large{f(x) = 4x^2+1}[/latex] and [latex]\large{g(x) = 3x-4}[/latex]

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

[latex]\large{f'(x) = 8x}[/latex] and [latex]\large{g'(x) = 3}[/latex]

Step 2:

The product rule formula states that:

[latex]\small{\frac{d}{dx}f(x)g(x) = f(x)g'(x) + f'(x)g(x)}[/latex]

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

[latex]\small{\frac{d}{dx}[(4x^2+1)(3x-4)]}[/latex]

[latex]\small{= (4x^2+1)(3) + (8x)(3x-4)}[/latex]

Step 3:

Simplifying, we have:

[latex]\small{ = 12x^2+3+24x^2-32x = 36x^2-32x+3}[/latex]

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

[latex]\large{\frac{dy}{dx} = 36x^2-32x+3}[/latex]

Solution Complete!