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

The derivative of (x^2+4x^{1/2}/(x^2)) is -6/(x^{5/2)}

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

The first derivative of (x^2+4x^{1/2}/(x^2)) is -6/(x^{5/2)}

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\frac{d}{d x} (\frac{x ^{2} + 4 x ^{\frac{1}{2}}}{x ^{2}})

- \frac{6}{x ^{\frac{5}{2}}}

Solution steps

\frac{d}{d x} (\frac{x ^{2} + 4 x ^{\frac{1}{2}}}{x ^{2}})

Apply the Quotient Rule:

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

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

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

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

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

Simplify

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

= - \frac{6}{x ^{\frac{5}{2}}}

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