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

To use the product rule, we also need the derivative of these functions. To find the derivative of the first function, we can use the chain rule, as follows:

[latex]\large{\frac{d}{dx}e^{3x} = e^{3x}\cdot\frac{d}{dx}3x = 3e^{3x}}[/latex]

Thus, [latex]\large{f'(x) = 3e^{3x}}[/latex]

and [latex]h'(x)[/latex] can easily be found to be:

[latex]\large{h'(x) = 2x }[/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]\large{\frac{d}{dx}((e^{3x})(x^2))}[/latex]

[latex] \large{= (e^{3x})(2x) + (3e^{3x})(x^2)}[/latex]

Step 3:

Simplifying, we have:

[latex]\large{ = 2xe^{3x} + 3x^2e^{3x}}[/latex]

Factoring out the common term, we have:

[latex]\large{ = xe^{3x}(2 + 3x)}[/latex]

This is the final answer for the derivative. Thus:

[latex]\large{g'(x) = xe^{3x}(2+3x)}[/latex]

Solution Complete!