What is the derivative of (e^{-x}(x+1)y) ?

The derivative of (e^{-x}(x+1)y) is -e^{-x}xy+e^{-x}x(dy)/(dx)+e^{-x}(dy)/(dx)

What is the first derivative of (e^{-x}(x+1)y) ?

The first derivative of (e^{-x}(x+1)y) is -e^{-x}xy+e^{-x}x(dy)/(dx)+e^{-x}(dy)/(dx)

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\frac{d}{d x} ((e^{- x} (x + 1)) y)

- e^{- x} x y + e^{- x} x \frac{d y}{d x} + e^{- x} \frac{d y}{d x}

Solution steps

\frac{d}{d x} ((e^{- x} (x + 1)) y)

Apply the Product Rule:

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

f = e^{- x}, g = (x + 1) y

= \frac{d}{d x} (e^{- x}) (x + 1) y + \frac{d}{d x} ((x + 1) y) e^{- x}

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

\frac{d}{d x} ((x + 1) y) = y + \frac{d y}{d x} (x + 1)

= (- e^{- x}) (x + 1) y + (y + \frac{d y}{d x} (x + 1)) e^{- x}

Simplify

(- e^{- x}) (x + 1) y + (y + \frac{d y}{d x} (x + 1)) e^{- x} : - e^{- x} x y + e^{- x} x \frac{d y}{d x} + e^{- x} \frac{d y}{d x}

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

- 6.5 \cdot 1 0^{- 4} (30)^{2}

\frac{3 ^{5}}{5 !} (e^{- 3})

6 y^{2} (2 y^{\circ})^{2}

6.62 \cdot 1 0^{- 16} \cdot 9.8049

What is the derivative of (e^{-x}(x+1)y) ?
The derivative of (e^{-x}(x+1)y) is -e^{-x}xy+e^{-x}x(dy)/(dx)+e^{-x}(dy)/(dx)
What is the first derivative of (e^{-x}(x+1)y) ?
The first derivative of (e^{-x}(x+1)y) is -e^{-x}xy+e^{-x}x(dy)/(dx)+e^{-x}(dy)/(dx)