What is the d/(dt)((e^{-t}-e^{-t}t)/(e^t)) ?

The d/(dt)((e^{-t}-e^{-t}t)/(e^t)) is (2t-3)/(e^{2t)}

What is the first d/(dt)((e^{-t}-e^{-t}t)/(e^t)) ?

The first d/(dt)((e^{-t}-e^{-t}t)/(e^t)) is (2t-3)/(e^{2t)}

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

\frac{2 t - 3}{e ^{2 t}}

Solution steps

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

Apply the Quotient Rule:

(\frac{f}{g})^{'} = \frac{f ^{'} \cdot g - g ^{'} \cdot f}{g ^{2}}

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

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

\frac{d}{d t} (e^{t}) = e^{t}

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

Simplify

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

= \frac{2 t - 3}{e ^{2 t}}

n = 0 \sum \infty n (\frac{2}{3})^{n}

\frac{d}{d x} (cos (x) - x sin (x))

f (x) = x^{2} e^{- x^{2}}

\int \frac{4 x + 1}{x ( x ^{2} - 2 x + 1 )} d x

x \cdot (1 - x)

What is the d/(dt)((e^{-t}-e^{-t}t)/(e^t)) ?
The d/(dt)((e^{-t}-e^{-t}t)/(e^t)) is (2t-3)/(e^{2t)}
What is the first d/(dt)((e^{-t}-e^{-t}t)/(e^t)) ?
The first d/(dt)((e^{-t}-e^{-t}t)/(e^t)) is (2t-3)/(e^{2t)}