What is the derivative of e^{-t/2}-(e^{-t/2}t)/2 ?

The derivative of e^{-t/2}-(e^{-t/2}t)/2 is (e^{-t/2}t)/4-e^{-t/2}

What is the first derivative of e^{-t/2}-(e^{-t/2}t)/2 ?

The first derivative of e^{-t/2}-(e^{-t/2}t)/2 is (e^{-t/2}t)/4-e^{-t/2}

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derivative of

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

\frac{e ^{- \frac{t}{2}} t}{4} - e^{- \frac{t}{2}}

Solution steps

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

Apply the Sum/Difference Rule:

(f \pm g)^{'} = f^{'} \pm g^{'}

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

\frac{d}{d t} (e^{- \frac{t}{2}}) = - \frac{1}{2} e^{- \frac{t}{2}}

\frac{d}{d t} (\frac{e ^{- \frac{t}{2}} t}{2}) = \frac{1}{2} (- \frac{e ^{- \frac{t}{2}} t}{2} + e^{- \frac{t}{2}})

= - \frac{1}{2} e^{- \frac{t}{2}} - \frac{1}{2} (- \frac{e ^{- \frac{t}{2}} t}{2} + e^{- \frac{t}{2}})

Simplify

- \frac{1}{2} e^{- \frac{t}{2}} - \frac{1}{2} (- \frac{e ^{- \frac{t}{2}} t}{2} + e^{- \frac{t}{2}}) : \frac{e ^{- \frac{t}{2}} t}{4} - e^{- \frac{t}{2}}

= \frac{e ^{- \frac{t}{2}} t}{4} - e^{- \frac{t}{2}}

What is the derivative of e^{-t/2}-(e^{-t/2}t)/2 ?
The derivative of e^{-t/2}-(e^{-t/2}t)/2 is (e^{-t/2}t)/4-e^{-t/2}
What is the first derivative of e^{-t/2}-(e^{-t/2}t)/2 ?
The first derivative of e^{-t/2}-(e^{-t/2}t)/2 is (e^{-t/2}t)/4-e^{-t/2}