What is the derivative of ((sqrt(x)^3)/((2x^7+x)^2)) ?

The derivative of ((sqrt(x)^3)/((2x^7+x)^2)) is -(50x^6+1)/(2x^{3/2)(2x^6+1)^3}

What is the first derivative of ((sqrt(x)^3)/((2x^7+x)^2)) ?

The first derivative of ((sqrt(x)^3)/((2x^7+x)^2)) is -(50x^6+1)/(2x^{3/2)(2x^6+1)^3}

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\frac{d}{d x} (\frac{( x ) ^{3}}{( 2 x ^{7} + x ) ^{2}})

- \frac{50 x ^{6} + 1}{2 x ^{\frac{3}{2}} ( 2 x ^{6} + 1 ) ^{3}}

Solution steps

\frac{d}{d x} (\frac{( x ) ^{3}}{( 2 x ^{7} + x ) ^{2}})

Apply the Quotient Rule:

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

= \frac{\frac{d}{d x} ( ( x ) ^{3} ) ( 2 x ^{7} + x ) ^{2} - \frac{d}{d x} ( ( 2 x ^{7} + x ) ^{2} ) ( x ) ^{3}}{( ( 2 x ^{7} + x ) ^{2} ) ^{2}}

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

\frac{d}{d x} ((2 x^{7} + x)^{2}) = 2 (2 x^{7} + x) (14 x^{6} + 1)

= \frac{\frac{3 x}{2} ( 2 x ^{7} + x ) ^{2} - 2 ( 2 x ^{7} + x ) ( 14 x ^{6} + 1 ) ( x ) ^{3}}{( ( 2 x ^{7} + x ) ^{2} ) ^{2}}

Simplify

\frac{\frac{3 x}{2} ( 2 x ^{7} + x ) ^{2} - 2 ( 2 x ^{7} + x ) ( 14 x ^{6} + 1 ) ( x ) ^{3}}{( ( 2 x ^{7} + x ) ^{2} ) ^{2}} : - \frac{50 x ^{6} + 1}{2 x ^{\frac{3}{2}} ( 2 x ^{6} + 1 ) ^{3}}

= - \frac{50 x ^{6} + 1}{2 x ^{\frac{3}{2}} ( 2 x ^{6} + 1 ) ^{3}}

What is the derivative of ((sqrt(x)^3)/((2x^7+x)^2)) ?
The derivative of ((sqrt(x)^3)/((2x^7+x)^2)) is -(50x^6+1)/(2x^{3/2)(2x^6+1)^3}
What is the first derivative of ((sqrt(x)^3)/((2x^7+x)^2)) ?
The first derivative of ((sqrt(x)^3)/((2x^7+x)^2)) is -(50x^6+1)/(2x^{3/2)(2x^6+1)^3}