What is the derivative of \sqrt[x]{e^1} ?

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

What is the first derivative of \sqrt[x]{e^1} ?

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

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

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

Solution steps

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

Apply exponent rule:

a^{b} = e^{b l n (a)}

x e^{1} = e^{\frac{1}{x} l n (e^{1})}

= \frac{d}{d x} (e^{\frac{1}{x} l n (e^{1})})

Apply the chain rule:

e^{\frac{1}{x} l n (e^{1})} \frac{d}{d x} (\frac{1}{x} ln (e^{1}))

= e^{\frac{1}{x} l n (e^{1})} \frac{d}{d x} (\frac{1}{x} ln (e^{1}))

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

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

Simplify

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

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

\int \frac{2}{x ^{2}} cos (\frac{2}{x}) d x

\frac{t ^{2}}{2}

\int 2 x x^{2} + 5 d x

\int \frac{3 x}{( 1 - 4 x ^{2} ) ^{2}} d x

x \to - 4 lim (2 x^{7} + 8 x^{4} - 1)

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