What is the derivative of (1/(x^2)-5/(x^4))(x+8x^3) ?

The derivative of (1/(x^2)-5/(x^4))(x+8x^3) is (39)/(x^2)+(15)/(x^4)+8

What is the first derivative of (1/(x^2)-5/(x^4))(x+8x^3) ?

The first derivative of (1/(x^2)-5/(x^4))(x+8x^3) is (39)/(x^2)+(15)/(x^4)+8

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derivative of

(\frac{1}{x ^{2}} - \frac{5}{x ^{4}}) (x + 8 x^{3})

\frac{39}{x ^{2}} + \frac{15}{x ^{4}} + 8

Solution steps

\frac{d}{d x} ((\frac{1}{x ^{2}} - \frac{5}{x ^{4}}) (x + 8 x^{3}))

Apply the Product Rule:

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

f = \frac{1}{x ^{2}} - \frac{5}{x ^{4}}, g = x + 8 x^{3}

= \frac{d}{d x} (\frac{1}{x ^{2}} - \frac{5}{x ^{4}}) (x + 8 x^{3}) + \frac{d}{d x} (x + 8 x^{3}) (\frac{1}{x ^{2}} - \frac{5}{x ^{4}})

\frac{d}{d x} (\frac{1}{x ^{2}} - \frac{5}{x ^{4}}) = - \frac{2}{x ^{3}} + \frac{20}{x ^{5}}

\frac{d}{d x} (x + 8 x^{3}) = 1 + 24 x^{2}

= (- \frac{2}{x ^{3}} + \frac{20}{x ^{5}}) (x + 8 x^{3}) + (1 + 24 x^{2}) (\frac{1}{x ^{2}} - \frac{5}{x ^{4}})

Simplify

(- \frac{2}{x ^{3}} + \frac{20}{x ^{5}}) (x + 8 x^{3}) + (1 + 24 x^{2}) (\frac{1}{x ^{2}} - \frac{5}{x ^{4}}) : \frac{39}{x ^{2}} + \frac{15}{x ^{4}} + 8

= \frac{39}{x ^{2}} + \frac{15}{x ^{4}} + 8

What is the derivative of (1/(x^2)-5/(x^4))(x+8x^3) ?
The derivative of (1/(x^2)-5/(x^4))(x+8x^3) is (39)/(x^2)+(15)/(x^4)+8
What is the first derivative of (1/(x^2)-5/(x^4))(x+8x^3) ?
The first derivative of (1/(x^2)-5/(x^4))(x+8x^3) is (39)/(x^2)+(15)/(x^4)+8