Final answer:
The derivative of the function f(x) = e^x cosh(x) is found using the product rule and is f'(x) = e^x(sinh(x) + cosh(x)).
Step-by-step explanation:
To find the derivative of the function f(x) = ex cosh(x), we must use the product rule, which states that the derivative of two functions multiplied together is the first function times the derivative of the second plus the second function times the derivative of the first. Knowing that the derivative of ex is ex and the derivative of cosh(x) is sinh(x), we apply the product rule.
So, the derivative f'(x) is:
Adding both terms, we get f'(x) = ex sinh(x) + cosh(x) ex
Simplify by factoring out ex, gives us f'(x) = ex(sinh(x) + cosh(x)), which is the final result.