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

The derivative of (x^{-2}+x^3(4x)^{-2}) is (3(1+x^3)^2(-2+x^3))/(16x^9)-((1+x^3)^3)/(8x^9)

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

The first derivative of (x^{-2}+x^3(4x)^{-2}) is (3(1+x^3)^2(-2+x^3))/(16x^9)-((1+x^3)^3)/(8x^9)

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

\frac{3 ( 1 + x ^{3} ) ^{2} ( - 2 + x ^{3} )}{16 x ^{9}} - \frac{( 1 + x ^{3} ) ^{3}}{8 x ^{9}}

Solution steps

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

Apply the Product Rule:

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

f = (x^{- 2} + x)^{3}, g = (4 x)^{- 2}

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

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

\frac{d}{d x} ((4 x)^{- 2}) = - \frac{1}{8 x ^{3}}

= 3 (\frac{1}{x ^{2}} + x)^{2} (- \frac{2}{x ^{3}} + 1) (4 x)^{- 2} + (- \frac{1}{8 x ^{3}}) (x^{- 2} + x)^{3}

Simplify

3 (\frac{1}{x ^{2}} + x)^{2} (- \frac{2}{x ^{3}} + 1) (4 x)^{- 2} + (- \frac{1}{8 x ^{3}}) (x^{- 2} + x)^{3} : \frac{3 ( 1 + x ^{3} ) ^{2} ( - 2 + x ^{3} )}{16 x ^{9}} - \frac{( 1 + x ^{3} ) ^{3}}{8 x ^{9}}

= \frac{3 ( 1 + x ^{3} ) ^{2} ( - 2 + x ^{3} )}{16 x ^{9}} - \frac{( 1 + x ^{3} ) ^{3}}{8 x ^{9}}

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

\int (1 - 5 y) d y

\frac{d}{d x} (arcsin (ln (x)))

\frac{1}{s ^{2} - 3 s + 2}

arctan (5 x^{2} - 1)

What is the derivative of (x^{-2}+x^3(4x)^{-2}) ?
The derivative of (x^{-2}+x^3(4x)^{-2}) is (3(1+x^3)^2(-2+x^3))/(16x^9)-((1+x^3)^3)/(8x^9)
What is the first derivative of (x^{-2}+x^3(4x)^{-2}) ?
The first derivative of (x^{-2}+x^3(4x)^{-2}) is (3(1+x^3)^2(-2+x^3))/(16x^9)-((1+x^3)^3)/(8x^9)