What is the derivative of (x^3)/(e^{1/x)} ?

The derivative of (x^3)/(e^{1/x)} is (x(3x+1))/(e^{1/x)}

What is the first derivative of (x^3)/(e^{1/x)} ?

The first derivative of (x^3)/(e^{1/x)} is (x(3x+1))/(e^{1/x)}

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derivative of

\frac{x ^{3}}{e ^{\frac{1}{x}}}

\frac{x ( 3 x + 1 )}{e ^{\frac{1}{x}}}

Solution steps

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

Apply the Quotient Rule:

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

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

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

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

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

Simplify

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

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

What is the derivative of (x^3)/(e^{1/x)} ?
The derivative of (x^3)/(e^{1/x)} is (x(3x+1))/(e^{1/x)}
What is the first derivative of (x^3)/(e^{1/x)} ?
The first derivative of (x^3)/(e^{1/x)} is (x(3x+1))/(e^{1/x)}