What is (\partial)/(\partial y)(sin(nx)e^{-ny}) ?

The answer to (\partial)/(\partial y)(sin(nx)e^{-ny}) is -ne^{-ny}sin(nx)

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(\partial)/(\partial y)(sin(nx)e^{-ny})

\frac{\partial}{\partial y} (sin (n x) e^{- n y})

- n e^{- n y} sin (n x)

Solution steps

\frac{\partial}{\partial y} (sin (n x) e^{- n y})

Treat

n, x

as a constant

Take the constant out:

(a \cdot f)^{'} = a \cdot f^{'}

= sin (n x) \frac{\partial}{\partial y} (e^{- n y})

Apply the chain rule:

e^{- n y} \frac{\partial}{\partial y} (- n y)

= e^{- n y} \frac{\partial}{\partial y} (- n y)

\frac{\partial}{\partial y} (- n y) = - n

= sin (n x) e^{- n y} (- n)

Simplify

= - n e^{- n y} sin (n x)

What is (\partial)/(\partial y)(sin(nx)e^{-ny}) ?
The answer to (\partial)/(\partial y)(sin(nx)e^{-ny}) is -ne^{-ny}sin(nx)