What is (\partial)/(\partial x)((x^2-1)/(x^2+x+1)) ?

The answer to (\partial)/(\partial x)((x^2-1)/(x^2+x+1)) is (x^2+4x+1)/((x^2+x+1)^2)

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(\partial)/(\partial x)((x^2-1)/(x^2+x+1))

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

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

Solution steps

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

Apply the Quotient Rule:

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

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

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

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

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

Expand

2 x (x^{2} + x + 1) - (2 x + 1) (x^{2} - 1) : x^{2} + 4 x + 1

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

\int 2^{r} d r

f (x) = 3 + x

x^{2} \cdot sec (\frac{1}{x})

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

\int \frac{1}{( 2 - 5 x ) ^{2}} d x

What is (\partial)/(\partial x)((x^2-1)/(x^2+x+1)) ?
The answer to (\partial)/(\partial x)((x^2-1)/(x^2+x+1)) is (x^2+4x+1)/((x^2+x+1)^2)