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{f(x) = e^x-x}[/latex] and [latex]\large{g(x) = \sin(x)}[/latex]

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

[latex]\large{f'(x) = e^x-1}[/latex] and [latex]\large{g'(x) = \cos(x)}[/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{e^x-x}{\sin(x)})}[/latex]

[latex]\large{ = \frac{(\sin(x))(e^x-1) \:- \:(e^x-x)(\cos(x))}{\sin^2(x)}}[/latex]

Step 3:

Simplifying, we have:

[latex]\Large{ = \frac{\sin(x)(e^x-1) – \cos(x)(e^x-x)}{\sin^2(x)}}[/latex]

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

[latex]\Large{h'(x) = \frac{\sin(x)(e^x-1) – \cos(x)(e^x-x)}{\sin^2(x)}}[/latex]

Solution Complete!