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`y = e^(alpha x) sin(beta x)` Find y' and y''

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Using the product rule:

d/dx f(x)g(x) = f'(x)g(x) + f(x)g'(x)

`y'= d/dx [e^(alpha x) sin (beta x)]`

`= e^(alpha x) * alpha * sin(beta x) + e^(alpha x) * cos(beta x)* beta`

`= e^(alpha x) *[alpha sin(beta x) + beta * cos (beta x)]`

`y'' = d/dx (e^(alpha x)) * [alpha sin (beta x) + beta cos (beta x)] `

`+ e^(alpha x) d/dx [alpha sin(betax) + beta cos(betax)]`

`= alpha e^(alphax) * [alpha sin(beta x)+ beta cos(betax)] `

`+e^(alpha x)*[alpha*beta cos(beta x)-beta^2 sin(betax)]`

`= e^(alpha x) *[(alpha^2-beta^2)sin(betax) + 2alphabeta cos(betax)]`

hope this helps.

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