What is the derivative of 1/(sqrt(e^x)) ?

The derivative of 1/(sqrt(e^x)) is -1/(2e^{x/2)}

What is the first derivative of 1/(sqrt(e^x)) ?

The first derivative of 1/(sqrt(e^x)) is -1/(2e^{x/2)}

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\frac{d}{d x} (\frac{1}{e ^{x}})

- \frac{1}{2 e ^{\frac{x}{2}}}

Solution steps

\frac{d}{d x} (\frac{1}{e ^{x}})

Apply exponent rule:

\frac{1}{a} = a^{- 1}

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

Apply the chain rule:

- \frac{1}{2 ( e ^{x} ) ^{\frac{3}{2}}} \frac{d}{d x} (e^{x})

= - \frac{1}{2 ( e ^{x} ) ^{\frac{3}{2}}} \frac{d}{d x} (e^{x})

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

= - \frac{1}{2 ( e ^{x} ) ^{\frac{3}{2}}} e^{x}

Simplify

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

= - \frac{1}{2 e ^{\frac{x}{2}}}

What is the derivative of 1/(sqrt(e^x)) ?
The derivative of 1/(sqrt(e^x)) is -1/(2e^{x/2)}
What is the first derivative of 1/(sqrt(e^x)) ?
The first derivative of 1/(sqrt(e^x)) is -1/(2e^{x/2)}