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

The derivative of ((x^2+x)/(x^2)) is -1/(x^2)

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

The first derivative of ((x^2+x)/(x^2)) is -1/(x^2)

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

- \frac{1}{x ^{2}}

Solution steps

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

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

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

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

Simplify

\frac{( 2 x + 1 ) x ^{2} - 2 x ( x ^{2} + x )}{( x ^{2} ) ^{2}} : - \frac{1}{x ^{2}}

= - \frac{1}{x ^{2}}

What is the derivative of ((x^2+x)/(x^2)) ?
The derivative of ((x^2+x)/(x^2)) is -1/(x^2)
What is the first derivative of ((x^2+x)/(x^2)) ?
The first derivative of ((x^2+x)/(x^2)) is -1/(x^2)