What is (\partial)/(\partial x)(t^3e^{-(x^2)/(2t)}) ?

The answer to (\partial)/(\partial x)(t^3e^{-(x^2)/(2t)}) is -e^{-(x^2)/(2t)}t^2x

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(\partial)/(\partial x)(t^3e^{-(x^2)/(2t)})

\frac{\partial}{\partial x} (t^{3} e^{- \frac{x ^{2}}{2 t}})

- e^{- \frac{x ^{2}}{2 t}} t^{2} x

Solution steps

\frac{\partial}{\partial x} (t^{3} e^{- \frac{x ^{2}}{2 t}})

Treat

t

as a constant

Take the constant out:

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

= t^{3} \frac{\partial}{\partial x} (e^{- \frac{x ^{2}}{2 t}})

Apply the chain rule:

e^{- \frac{x ^{2}}{2 t}} \frac{\partial}{\partial x} (- \frac{x ^{2}}{2 t})

= e^{- \frac{x ^{2}}{2 t}} \frac{\partial}{\partial x} (- \frac{x ^{2}}{2 t})

\frac{\partial}{\partial x} (- \frac{x ^{2}}{2 t}) = - \frac{x}{t}

= t^{3} e^{- \frac{x ^{2}}{2 t}} (- \frac{x}{t})

Simplify

t^{3} e^{- \frac{x ^{2}}{2 t}} (- \frac{x}{t}) : - e^{- \frac{x ^{2}}{2 t}} t^{2} x

= - e^{- \frac{x ^{2}}{2 t}} t^{2} x

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What is (\partial)/(\partial x)(t^3e^{-(x^2)/(2t)}) ?
The answer to (\partial)/(\partial x)(t^3e^{-(x^2)/(2t)}) is -e^{-(x^2)/(2t)}t^2x