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

To use the product rule, we also need the derivative of these functions. This should be easy for us to do using our knowledge of common derivatives and the power rule. Thus:

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

[latex] \small{= (\sec(x))(2x) + (\sec(x)\tan(x))(x^2)}[/latex]

Step 3:

Simplifying, we have:

[latex]\large{ = 2x\sec(x) + x^2\sec(x)\tan(x)}[/latex]

Now, factoring out the common factor from both terms, we have:

[latex]\large{ = x\sec(x)(2 + x\tan(x))}[/latex]

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

[latex]\large{f'(x) = x\sec(x)(2 + x\tan(x))}[/latex]

Solution Complete!