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

The derivative of (x(x^2-1)/(sqrt(x^2+1))) is (2x^4+3x^2-1)/((x^2+1)sqrt(x^2+1))

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

The first derivative of (x(x^2-1)/(sqrt(x^2+1))) is (2x^4+3x^2-1)/((x^2+1)sqrt(x^2+1))

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

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

Solution steps

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

Apply the Quotient Rule:

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

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

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

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

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

Simplify

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

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

What is the derivative of (x(x^2-1)/(sqrt(x^2+1))) ?
The derivative of (x(x^2-1)/(sqrt(x^2+1))) is (2x^4+3x^2-1)/((x^2+1)sqrt(x^2+1))
What is the first derivative of (x(x^2-1)/(sqrt(x^2+1))) ?
The first derivative of (x(x^2-1)/(sqrt(x^2+1))) is (2x^4+3x^2-1)/((x^2+1)sqrt(x^2+1))