What is the derivative of (x^{1/3}-4)/(x^{7/2)} ?

The derivative of (x^{1/3}-4)/(x^{7/2)} is (-19x^{17/6}+84x^{5/2})/(6x^7)

What is the first derivative of (x^{1/3}-4)/(x^{7/2)} ?

The first derivative of (x^{1/3}-4)/(x^{7/2)} is (-19x^{17/6}+84x^{5/2})/(6x^7)

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derivative of

\frac{x ^{\frac{1}{3}} - 4}{x ^{\frac{7}{2}}}

\frac{- 19 x ^{\frac{17}{6}} + 84 x ^{\frac{5}{2}}}{6 x ^{7}}

Solution steps

\frac{d}{d x} (\frac{x ^{\frac{1}{3}} - 4}{x ^{\frac{7}{2}}})

Apply the Quotient Rule:

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

= \frac{\frac{d}{d x} ( x ^{\frac{1}{3}} - 4 ) x ^{\frac{7}{2}} - \frac{d}{d x} ( x ^{\frac{7}{2}} ) ( x ^{\frac{1}{3}} - 4 )}{( x ^{\frac{7}{2}} ) ^{2}}

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

\frac{d}{d x} (x^{\frac{7}{2}}) = \frac{7 x ^{\frac{5}{2}}}{2}

= \frac{\frac{1}{3 x ^{\frac{2}{3}}} x ^{\frac{7}{2}} - \frac{7 x ^{\frac{5}{2}}}{2} ( x ^{\frac{1}{3}} - 4 )}{( x ^{\frac{7}{2}} ) ^{2}}

Simplify

\frac{\frac{1}{3 x ^{\frac{2}{3}}} x ^{\frac{7}{2}} - \frac{7 x ^{\frac{5}{2}}}{2} ( x ^{\frac{1}{3}} - 4 )}{( x ^{\frac{7}{2}} ) ^{2}} : \frac{- 19 x ^{\frac{17}{6}} + 84 x ^{\frac{5}{2}}}{6 x ^{7}}

= \frac{- 19 x ^{\frac{17}{6}} + 84 x ^{\frac{5}{2}}}{6 x ^{7}}

y^{^{''}} + 10 y^{^{'}} + 25 y = - 50 e^{- 5 x}

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What is the derivative of (x^{1/3}-4)/(x^{7/2)} ?
The derivative of (x^{1/3}-4)/(x^{7/2)} is (-19x^{17/6}+84x^{5/2})/(6x^7)
What is the first derivative of (x^{1/3}-4)/(x^{7/2)} ?
The first derivative of (x^{1/3}-4)/(x^{7/2)} is (-19x^{17/6}+84x^{5/2})/(6x^7)