What is (\partial)/(\partial t)((w)(t^22te^{-t})) ?

The answer to (\partial)/(\partial t)((w)(t^22te^{-t})) is 2w(-e^{-t}t^3+3e^{-t}t^2)

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(\partial)/(\partial t)((w)(t^22te^{-t}))

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

2 w (- e^{- t} t^{3} + 3 e^{- t} t^{2})

Solution steps

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

Simplify

(w) (t^{2} \cdot 2 t e^{- t}) : 2 e^{- t} w t^{3}

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

Treat

w

as a constant

Take the constant out:

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

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

Apply the Product Rule:

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

f = e^{- t}, g = t^{3}

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

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

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

= 2 w ((- e^{- t}) t^{3} + 3 t^{2} e^{- t})

Simplify

= 2 w (- e^{- t} t^{3} + 3 e^{- t} t^{2})

e^{- (\frac{100}{100})^{3}}

\frac{1}{2 π 4.32} \cdot e^{- 0.746}

e^{- t}

462.43 (1 0^{- 3})

e^{- a t} \cdot u (t)

What is (\partial)/(\partial t)((w)(t^22te^{-t})) ?
The answer to (\partial)/(\partial t)((w)(t^22te^{-t})) is 2w(-e^{-t}t^3+3e^{-t}t^2)