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

The derivative of (-e^{-x/2}x+e^{-x/2}/2) is (e^{-x/2}x-3e^{-x/2})/4

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

The first derivative of (-e^{-x/2}x+e^{-x/2}/2) is (e^{-x/2}x-3e^{-x/2})/4

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

\frac{e ^{- \frac{x}{2}} x - 3 e ^{- \frac{x}{2}}}{4}

Solution steps

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

Take the constant out:

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

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

Apply the Sum/Difference Rule:

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

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

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

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

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

Simplify

\frac{1}{2} (- (- \frac{e ^{- \frac{x}{2}} x}{2} + e^{- \frac{x}{2}}) - \frac{1}{2} e^{- \frac{x}{2}}) : \frac{e ^{- \frac{x}{2}} x - 3 e ^{- \frac{x}{2}}}{4}

= \frac{e ^{- \frac{x}{2}} x - 3 e ^{- \frac{x}{2}}}{4}

What is the derivative of (-e^{-x/2}x+e^{-x/2}/2) ?
The derivative of (-e^{-x/2}x+e^{-x/2}/2) is (e^{-x/2}x-3e^{-x/2})/4
What is the first derivative of (-e^{-x/2}x+e^{-x/2}/2) ?
The first derivative of (-e^{-x/2}x+e^{-x/2}/2) is (e^{-x/2}x-3e^{-x/2})/4