# Find the derivative. Simplify where possible.f(x) = x sinh x − 9 cosh x

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You need to use definitions of hyperbolic functions such that:

`sinh x = (e^x - e^(-x))/2`

`cosh x = (e^x+ e^(-x))/2`

You need to substitute `(e^x - e^(-x))/2` for`sinh x` and `(e^x+ e^(-x))/2` for `cosh x` in equation of function such that:

`f(x) = x*(e^x - e^(-x))/2 - 9 (e^x + e^(-x))/2`

You nee to differentiate with respect to x such that:

`f'(x) = (x*(e^x - e^(-x))/2)' - 9 ((e^x + e^(-x))/2)'`

`f'(x) = x'*(e^x - e^(-x))/2 + (1/2)*x*((e^x - e^(-x)))' - (9/2)*(e^x + e^(-x))'`

`f'(x) = (e^x - e^(-x))/2 + (1/2)*x*(e^x+ e^(-x)) - (9/2)*(e^x- e^(-x))`

`f'(x) = (1/2)*x*(e^x + e^(-x)) - 7(e^x - e^(-x))/2`

You need to factor out `1/2` such that:

`f'(x) = (1/2)*(xe^x + xe^(-x) - 7e^x + 7e^(-x))`

You need to factor out `e^x` and `e^(-x)` such that:

`f'(x) = (1/2)*(e^x(x-7) + e^(-x)(x + 7))`

**Hence, evaluating derivative of function `f(x) = x sinh x − 9 cosh x` using definitions of hyperbolic functions yields `f'(x) = (1/2)*(e^x(x-7) + e^(-x)(x + 7)).` **