What is the derivative of f(x)=(x^2+x-2)/(x^3+6) ?

The derivative of f(x)=(x^2+x-2)/(x^3+6) is (-x^4-2x^3+6x^2+12x+6)/((x^3+6)^2)

What is the first derivative of f(x)=(x^2+x-2)/(x^3+6) ?

The first derivative of f(x)=(x^2+x-2)/(x^3+6) is (-x^4-2x^3+6x^2+12x+6)/((x^3+6)^2)

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derivative of

f (x) = \frac{x ^{2} + x - 2}{x ^{3} + 6}

\frac{- x ^{4} - 2 x ^{3} + 6 x ^{2} + 12 x + 6}{( x ^{3} + 6 ) ^{2}}

Solution steps

\frac{d}{d x} (\frac{x ^{2} + x - 2}{x ^{3} + 6})

Apply the Quotient Rule:

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

= \frac{\frac{d}{d x} ( x ^{2} + x - 2 ) ( x ^{3} + 6 ) - \frac{d}{d x} ( x ^{3} + 6 ) ( x ^{2} + x - 2 )}{( x ^{3} + 6 ) ^{2}}

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

\frac{d}{d x} (x^{3} + 6) = 3 x^{2}

= \frac{( 2 x + 1 ) ( x ^{3} + 6 ) - 3 x ^{2} ( x ^{2} + x - 2 )}{( x ^{3} + 6 ) ^{2}}

Expand

(2 x + 1) (x^{3} + 6) - 3 x^{2} (x^{2} + x - 2) : - x^{4} - 2 x^{3} + 6 x^{2} + 12 x + 6

= \frac{- x ^{4} - 2 x ^{3} + 6 x ^{2} + 12 x + 6}{( x ^{3} + 6 ) ^{2}}

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

\frac{d}{d x} (ln (e^{x} - e^{a}))

\frac{d}{d x} (e^{x^{2} + 9 x})

\int sec^{3} (x) tan^{5} (x) d x

\int 2 x^{3} d x

What is the derivative of f(x)=(x^2+x-2)/(x^3+6) ?
The derivative of f(x)=(x^2+x-2)/(x^3+6) is (-x^4-2x^3+6x^2+12x+6)/((x^3+6)^2)
What is the first derivative of f(x)=(x^2+x-2)/(x^3+6) ?
The first derivative of f(x)=(x^2+x-2)/(x^3+6) is (-x^4-2x^3+6x^2+12x+6)/((x^3+6)^2)