Differentiate (cosh x - cos x) / (sinh x - sin x) with respect to x.

Steps to Solve

1. Identify the function and the quotient rule
   We begin with the function ( f(x) = \frac{\cosh x - \cos x}{\sinh x - \sin x} ).
   To find the derivative, we will use the quotient rule, which states that if ( f(x) = \frac{g(x)}{h(x)} ), then
   $$ f'(x) = \frac{g'(x)h(x) - g(x)h'(x)}{(h(x))^2} $$
   where ( g(x) = \cosh x - \cos x ) and ( h(x) = \sinh x - \sin x ).

2. Find the derivatives of g(x) and h(x)
   We need to find ( g'(x) ) and ( h'(x) ).

For ( g(x) = \cosh x - \cos x ):
The derivative ( g'(x) = \sinh x + \sin x ).

For ( h(x) = \sinh x - \sin x ):
The derivative ( h'(x) = \cosh x - \cos x ).

3. Plug into the quotient rule
   Now substitute ( g(x) ), ( g'(x) ), ( h(x) ), and ( h'(x) ) into the quotient rule formula.
   Thus, we have:

$$ f'(x) = \frac{(\sinh x + \sin x)(\sinh x - \sin x) - (\cosh x - \cos x)(\cosh x - \cos x)}{(\sinh x - \sin x)^2} $$

4. Simplify the expression
   Next, simplify the numerator:
   Expanding using the difference of squares gives:
   $$ (\sinh^2 x - \sin^2 x) - (\cosh^2 x - 2\cosh x \cos x + \cos^2 x) $$
   Combine terms, remembering that ( \cosh^2 x - \sinh^2 x = 1 ), leading to:
   $$ = (1 - \sin^2 x - \cos^2 x + 2\cosh x \cos x) $$

5. Final derivative expression
   Thus, the final derivative is:

$$ f'(x) = \frac{1 - \sin^2 x - \cos^2 x + 2 \cosh x \cos x}{(\sinh x - \sin x)^2} $$
Since ( \sin^2 x + \cos^2 x = 1 ), we can simplify it to:
$$ f'(x) = \frac{2 \cosh x \cos x}{(\sinh x - \sin x)^2} $$