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) = x}[/latex] and [latex]\large{g(x) = \tan(x)}[/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) = 1}[/latex] and [latex]\large{g'(x) = \sec^2(x)}[/latex]

Note that the derivative of tan(x) is a common derivative which you should remember!

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}(x\tan(x))}[/latex]

[latex]\large{ = (x)(\sec^2(x)) + (1)(\tan(x))}[/latex]

Step 3:

Simplifying, we have:

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

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

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

Solution Complete!