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Popular Calculus >

(d^2)/(dx^2)(xye^{(-x+3y)})

Solution

\frac{d ^{2}}{d x ^{2}} (x y e^{(- x + 3 y)})

Solution

- 2 e^{- x + 3 y} y + 6 e^{- x + 3 y} y \frac{d y}{d x} + 2 e^{- x + 3 y} \frac{d y}{d x} + e^{- x + 3 y} x y + 6 e^{- x + 3 y} x \frac{d y}{d x}^{2} + 9 e^{- x + 3 y} x y \frac{d y}{d x}^{2} - 2 e^{- x + 3 y} x \frac{d y}{d x} - 6 e^{- x + 3 y} x y \frac{d y}{d x} + e^{- x + 3 y} x \frac{d ^{2} y}{d x ^{2}} + 3 e^{- x + 3 y} x y \frac{d}{d x} (\frac{d y}{d x})

Solution steps

\frac{d ^{2}}{d x ^{2}} (x y e^{(- x + 3 y)})

\frac{d}{d x} (x y e^{- x + 3 y}) = e^{- x + 3 y} y + x (e^{- x + 3 y} \frac{d y}{d x} + e^{- x + 3 y} (- 1 + 3 \frac{d y}{d x}) y)

= \frac{d}{d x} (e^{- x + 3 y} y + x (e^{- x + 3 y} \frac{d y}{d x} + e^{- x + 3 y} (- 1 + 3 \frac{d y}{d x}) y))

\frac{d}{d x} (e^{- x + 3 y} y + x (e^{- x + 3 y} \frac{d y}{d x} + e^{- x + 3 y} (- 1 + 3 \frac{d y}{d x}) y)) = - 2 e^{- x + 3 y} y + 6 e^{- x + 3 y} y \frac{d y}{d x} + 2 e^{- x + 3 y} \frac{d y}{d x} + e^{- x + 3 y} x y + 6 e^{- x + 3 y} x \frac{d y}{d x}^{2} + 9 e^{- x + 3 y} x y \frac{d y}{d x}^{2} - 2 e^{- x + 3 y} x \frac{d y}{d x} - 6 e^{- x + 3 y} x y \frac{d y}{d x} + e^{- x + 3 y} x \frac{d ^{2} y}{d x ^{2}} + 3 e^{- x + 3 y} x y \frac{d}{d x} (\frac{d y}{d x})

= - 2 e^{- x + 3 y} y + 6 e^{- x + 3 y} y \frac{d y}{d x} + 2 e^{- x + 3 y} \frac{d y}{d x} + e^{- x + 3 y} x y + 6 e^{- x + 3 y} x \frac{d y}{d x}^{2} + 9 e^{- x + 3 y} x y \frac{d y}{d x}^{2} - 2 e^{- x + 3 y} x \frac{d y}{d x} - 6 e^{- x + 3 y} x y \frac{d y}{d x} + e^{- x + 3 y} x \frac{d ^{2} y}{d x ^{2}} + 3 e^{- x + 3 y} x y \frac{d}{d x} (\frac{d y}{d x})

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Frequently Asked Questions (FAQ)

What is (d^2)/(dx^2)(xye^{(-x+3y)}) ?
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