What is the derivative of (x^{-2}+x^{-3}(x^5-9x^2)) ?

The derivative of (x^{-2}+x^{-3}(x^5-9x^2)) is 3x^2+2x+9/(x^2)

What is the first derivative of (x^{-2}+x^{-3}(x^5-9x^2)) ?

The first derivative of (x^{-2}+x^{-3}(x^5-9x^2)) is 3x^2+2x+9/(x^2)

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\frac{d}{d x} ((x^{- 2} + x^{- 3}) (x^{5} - 9 x^{2}))

3 x^{2} + 2 x + \frac{9}{x ^{2}}

Solution steps

\frac{d}{d x} ((x^{- 2} + x^{- 3}) (x^{5} - 9 x^{2}))

Apply the Product Rule:

(f \cdot g)^{'} = f^{'} \cdot g + f \cdot g^{'}

f = x^{- 2} + x^{- 3}, g = x^{5} - 9 x^{2}

= \frac{d}{d x} (x^{- 2} + x^{- 3}) (x^{5} - 9 x^{2}) + \frac{d}{d x} (x^{5} - 9 x^{2}) (x^{- 2} + x^{- 3})

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

\frac{d}{d x} (x^{5} - 9 x^{2}) = 5 x^{4} - 18 x

= (- \frac{2}{x ^{3}} - \frac{3}{x ^{4}}) (x^{5} - 9 x^{2}) + (5 x^{4} - 18 x) (x^{- 2} + x^{- 3})

Simplify

(- \frac{2}{x ^{3}} - \frac{3}{x ^{4}}) (x^{5} - 9 x^{2}) + (5 x^{4} - 18 x) (x^{- 2} + x^{- 3}) : 3 x^{2} + 2 x + \frac{9}{x ^{2}}

= 3 x^{2} + 2 x + \frac{9}{x ^{2}}

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

\int 3 sin^{3} (3 x) cos^{3} (3 x) d x

sin (5 x) \cdot sin (4 x)

\int (x^{2} + x)^{12} (2 x + 1) d x

\int \frac{1}{2} sin (2 x) - cos (x) + C d x

What is the derivative of (x^{-2}+x^{-3}(x^5-9x^2)) ?
The derivative of (x^{-2}+x^{-3}(x^5-9x^2)) is 3x^2+2x+9/(x^2)
What is the first derivative of (x^{-2}+x^{-3}(x^5-9x^2)) ?
The first derivative of (x^{-2}+x^{-3}(x^5-9x^2)) is 3x^2+2x+9/(x^2)