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

The derivative of (x^2(x^3-1)/(sqrt(x^2+1))) is (4x^6+5x^4-x^3-2x)/((x^2+1)sqrt(x^2+1))

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

The first derivative of (x^2(x^3-1)/(sqrt(x^2+1))) is (4x^6+5x^4-x^3-2x)/((x^2+1)sqrt(x^2+1))

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

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

Solution steps

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

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

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

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

Simplify

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

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

x \to 5 lim (3 x^{2} - 4 x - 1)

f (x) = 8 x + x^{2} + 7 x + 3

\int_{1}^{2} y^{3} + \frac{1}{4 y ^{3}} d y

ln (1 + 4 x^{2} + 4 x)

\frac{\partial}{\partial x} (e^{- x} sin (2 y))

What is the derivative of (x^2(x^3-1)/(sqrt(x^2+1))) ?
The derivative of (x^2(x^3-1)/(sqrt(x^2+1))) is (4x^6+5x^4-x^3-2x)/((x^2+1)sqrt(x^2+1))
What is the first derivative of (x^2(x^3-1)/(sqrt(x^2+1))) ?
The first derivative of (x^2(x^3-1)/(sqrt(x^2+1))) is (4x^6+5x^4-x^3-2x)/((x^2+1)sqrt(x^2+1))