Differention (could you check my solutions please)

[enosthapa]

Homework Statement

Differentiate y=e^-xsin(3x)

2.The attempt at a solution

With Product rule
y'=-e^-xSin(3x)+3e^-xCos(3x)

And Logarithm way
Taking log both sides

lny= lne^-x+ln(Sin3x)

lny= 1/x+ln(Sin3x)

1/y*dy/dx=-x^-2+3(Cos3x/Sin3x)

y'=-(e^-xSin3x)/x²+3e^-xCos3x

 

[cragar]

your product rule one looks good , but I don't think log one is .
why do you go from ln(e^-x) to 1/x
dervative of ln(e^(-x)) is -1

 

[enosthapa]

your product rule one looks good , but I don't think log one is .
why do you go from ln(e^-x) to 1/x
dervative of ln(e^(-x)) is -1

I would make sense but this is what i did

ln(e^-x) = 1/(lne^x)
=1/(x.lne)
=1/x

 

[cragar]

ok but you see the difference ln(e^-x) = ln(1/(e^x))

 

[enosthapa]

Oh yes thankx... but can I do
ln(e^-x) = -x lne =-x

 

[SammyS]

Oh yes thankx... but can I do
ln(e^-x) = -x lne =-x

Of course!

ln(t) is the inverse function of e^(t), so ln(e^t) = t .