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

The answer to (\partial)/(\partial x)(sqrt((2x)/(x^2+1))) is (-x^2+1)/(sqrt(2)\sqrt{x)(x^2+1)^{3/2}}

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

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

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

Solution steps

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

Apply the chain rule:

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

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

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

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

Simplify

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

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

12 x^{2} + 3 x

f (x) = 4 - 5 x^{2}

x \to a lim (x - a + c)

n = 1 \sum \infty \frac{1}{5 ^{n}}

\frac{1}{x} \cdot \frac{d y}{d x} + 2 y - e^{- x^{2}} = 0, y (1) = \frac{2}{e}

What is (\partial)/(\partial x)(sqrt((2x)/(x^2+1))) ?
The answer to (\partial)/(\partial x)(sqrt((2x)/(x^2+1))) is (-x^2+1)/(sqrt(2)\sqrt{x)(x^2+1)^{3/2}}