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

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sllysmith | Student, Undergraduate | eNoter

Posted March 17, 2012 at 4:03 AM via web

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Find the derivative. Simplify where possible.

f(x) = x sinh x − 9 cosh x

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sciencesolve | Teacher | (Level 3) Educator Emeritus

Posted March 17, 2012 at 4:41 AM (Answer #1)

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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)).`

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