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

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

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

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

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

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

Solution steps

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

Apply the Product Rule:

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

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

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

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

Simplify

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

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

\int cos^{5} (2 y) d y

\frac{d}{d x} (cos (x^{- 3}))

(- 4, - 4), (2, 5)

f (x) = x, g (x) = x

f^{^{'}} (x) = \frac{9 x - 2}{8 x + 7}

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