Find derivatives for the following function e^(5x) * sinh3x + (4x+6)/(x^2+3x+5)

You need to use the product law to differentiate the product `e^(5x) * sinh3x`  with respect to x and you need to use quotient law to differentiate the fraction `(4x+6)/(x^2+3x+5)`  with respect to x such that:

`(e^(5x) * sinh3x)' = (e^(5x))' * sinh3x + e^(5x) * (sinh3x)'`

You need to remember what `sinh 3x`  means such that:

`sinh 3x = (e^(3x) - e^(-3x))/2`

`(sinh 3x)' = ((e^(3x) - e^(-3x))')/2`

`(sinh 3x)' = (3(e^(3x)+ e^(-3x)))/2`

`(sinh 3x)' = 3 cosh 3x`

`(e^(5x) * sinh3x)' = 5e^(5x)*sinh 3x + 3e^(5x)*cosh 3x`

`(e^(5x) * sinh3x)' = e^(5x)*(5sinh 3x + 3cosh 3x)`

You need to differentiate the fraction with respect to x such that:

`((4x+6)/(x^2+3x+5))' =((4x+6)'*(x^2+3x+5) -(4x+6)*(x^2+3x+5)')/((x^2+3x+5)^2)`

`((4x+6)/(x^2+3x+5))' = (4*(x^2+3x+5) - (4x+6)*(2x+3))/((x^2+3x+5)^2)`

`((4x+6)/(x^2+3x+5))' = (4x^2 + 12x + 20 - 8x^2 - 24x - 18)/((x^2+3x+5)^2)`

`((4x+6)/(x^2+3x+5))' = (-4x^2 - 12x + 2)/((x^2+3x+5)^2)`

`((4x+6)/(x^2+3x+5))' = -2(x^2 + 6x - 1)/((x^2+3x+5)^2)`

Hence, differentiating the function with respect to x yields `(e^(5x) * sinh3x + (4x+6)/(x^2+3x+5))' =e^(5x)*(5sinh 3x + 3cosh 3x) - 2(x^2 + 6x - 1)/((x^2+3x+5)^2).`

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