What is the derivative of 1/(x(x^2+e^x)) ?

The derivative of 1/(x(x^2+e^x)) is -(3x^2+e^xx+e^x)/(x^2(x^2+e^x)^2)

What is the first derivative of 1/(x(x^2+e^x)) ?

The first derivative of 1/(x(x^2+e^x)) is -(3x^2+e^xx+e^x)/(x^2(x^2+e^x)^2)

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derivative of

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

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

Solution steps

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

Apply exponent rule:

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

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

Apply the chain rule:

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

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

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

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

Simplify

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

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

What is the derivative of 1/(x(x^2+e^x)) ?
The derivative of 1/(x(x^2+e^x)) is -(3x^2+e^xx+e^x)/(x^2(x^2+e^x)^2)
What is the first derivative of 1/(x(x^2+e^x)) ?
The first derivative of 1/(x(x^2+e^x)) is -(3x^2+e^xx+e^x)/(x^2(x^2+e^x)^2)