What is the derivative of e^{-x}sin(y) ?

The derivative of e^{-x}sin(y) is -e^{-x}sin(y)+e^{-x}cos(y)(dy)/(dx)

What is the first derivative of e^{-x}sin(y) ?

The first derivative of e^{-x}sin(y) is -e^{-x}sin(y)+e^{-x}cos(y)(dy)/(dx)

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

- e^{- x} sin (y) + e^{- x} cos (y) \frac{d y}{d x}

Solution steps

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

Apply the Product Rule:

(f \cdot g)^{'} = f^{'} \cdot g + f \cdot g^{'}

f = e^{- x}, g = sin (y)

= \frac{d}{d x} (e^{- x}) sin (y) + \frac{d}{d x} (sin (y)) e^{- x}

\frac{d}{d x} (e^{- x}) = - e^{- x}

\frac{d}{d x} (sin (y)) = cos (y) \frac{d y}{d x}

= (- e^{- x}) sin (y) + cos (y) \frac{d y}{d x} e^{- x}

Simplify

= - e^{- x} sin (y) + e^{- x} cos (y) \frac{d y}{d x}

\frac{d}{d x} (e^{- 18 x})

\frac{\partial}{\partial x} (4 + x^{2})

\frac{d}{d x} (e^{- x} - e^{x} x)

\frac{d}{d x} (x^{4} + 4 e^{x})

0 \to 1 lim (e^{0})

What is the derivative of e^{-x}sin(y) ?
The derivative of e^{-x}sin(y) is -e^{-x}sin(y)+e^{-x}cos(y)(dy)/(dx)
What is the first derivative of e^{-x}sin(y) ?
The first derivative of e^{-x}sin(y) is -e^{-x}sin(y)+e^{-x}cos(y)(dy)/(dx)