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

The derivative of-(e^{-x}(x^3+3(x^2+2x+2))/(x^4)) is (e^{-x}(x^4+4(x^3+3(x^2+2x+2))))/(x^5)

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

The first derivative of-(e^{-x}(x^3+3(x^2+2x+2))/(x^4)) is (e^{-x}(x^4+4(x^3+3(x^2+2x+2))))/(x^5)

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

\frac{e ^{- x} ( x ^{4} + 4 ( x ^{3} + 3 ( x ^{2} + 2 x + 2 ) ) )}{x ^{5}}

Solution steps

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

Take the constant out:

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

= - \frac{d}{d x} (\frac{e ^{- x} ( x ^{3} + 3 ( x ^{2} + 2 x + 2 ) )}{x ^{4}})

Apply the Quotient Rule:

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

= - \frac{\frac{d}{d x} ( e ^{- x} ( x ^{3} + 3 ( x ^{2} + 2 x + 2 ) ) ) x ^{4} - \frac{d}{d x} ( x ^{4} ) e ^{- x} ( x ^{3} + 3 ( x ^{2} + 2 x + 2 ) )}{( x ^{4} ) ^{2}}

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

\frac{d}{d x} (x^{4}) = 4 x^{3}

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

Simplify

- \frac{( - e ^{- x} x ^{3} ) x ^{4} - 4 x ^{3} e ^{- x} ( x ^{3} + 3 ( x ^{2} + 2 x + 2 ) )}{( x ^{4} ) ^{2}} : \frac{e ^{- x} ( x ^{4} + 4 ( x ^{3} + 3 ( x ^{2} + 2 x + 2 ) ) )}{x ^{5}}

= \frac{e ^{- x} ( x ^{4} + 4 ( x ^{3} + 3 ( x ^{2} + 2 x + 2 ) ) )}{x ^{5}}

f (x) = \frac{x ^{2} + 8}{x + 28}, (5, 1)

(4, - 9), (2, - 2)

\int x^{m - 1} d x

y^{^{'}} = \frac{1}{1 - x ^{2}}

\int y e^{3 y^{2}} d y

What is the derivative of-(e^{-x}(x^3+3(x^2+2x+2))/(x^4)) ?
The derivative of-(e^{-x}(x^3+3(x^2+2x+2))/(x^4)) is (e^{-x}(x^4+4(x^3+3(x^2+2x+2))))/(x^5)
What is the first derivative of-(e^{-x}(x^3+3(x^2+2x+2))/(x^4)) ?
The first derivative of-(e^{-x}(x^3+3(x^2+2x+2))/(x^4)) is (e^{-x}(x^4+4(x^3+3(x^2+2x+2))))/(x^5)