What is the derivative of sqrt(x/(x^2+5)) ?

The derivative of sqrt(x/(x^2+5)) is (-x^2+5)/(2sqrt(x)(x^2+5)^{3/2)}

What is the first derivative of sqrt(x/(x^2+5)) ?

The first derivative of sqrt(x/(x^2+5)) is (-x^2+5)/(2sqrt(x)(x^2+5)^{3/2)}

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\frac{d}{d x} (\frac{x}{x ^{2} + 5})

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

Solution steps

\frac{d}{d x} (\frac{x}{x ^{2} + 5})

Apply the chain rule:

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

= \frac{1}{2 \frac{x}{x ^{2} + 5}} \frac{d}{d x} (\frac{x}{x ^{2} + 5})

\frac{d}{d x} (\frac{x}{x ^{2} + 5}) = \frac{- x ^{2} + 5}{( x ^{2} + 5 ) ^{2}}

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

Simplify

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

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

What is the derivative of sqrt(x/(x^2+5)) ?
The derivative of sqrt(x/(x^2+5)) is (-x^2+5)/(2sqrt(x)(x^2+5)^{3/2)}
What is the first derivative of sqrt(x/(x^2+5)) ?
The first derivative of sqrt(x/(x^2+5)) is (-x^2+5)/(2sqrt(x)(x^2+5)^{3/2)}