What is the derivative of sqrt((5x+1/(2x^2+1))) ?

The derivative of sqrt((5x+1/(2x^2+1))) is (-10x^2-4x+5)/(2sqrt(5x+1)(2x^2+1)^{3/2)}

What is the first derivative of sqrt((5x+1/(2x^2+1))) ?

The first derivative of sqrt((5x+1/(2x^2+1))) is (-10x^2-4x+5)/(2sqrt(5x+1)(2x^2+1)^{3/2)}

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

\frac{- 10 x ^{2} - 4 x + 5}{2 5 x + 1 ( 2 x ^{2} + 1 ) ^{\frac{3}{2}}}

Solution steps

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

Apply the chain rule:

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

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

\frac{d}{d x} (\frac{5 x + 1}{2 x ^{2} + 1}) = \frac{- 10 x ^{2} - 4 x + 5}{( 2 x ^{2} + 1 ) ^{2}}

= \frac{1}{2 \frac{5 x + 1}{2 x ^{2} + 1}} \cdot \frac{- 10 x ^{2} - 4 x + 5}{( 2 x ^{2} + 1 ) ^{2}}

Simplify

\frac{1}{2 \frac{5 x + 1}{2 x ^{2} + 1}} \cdot \frac{- 10 x ^{2} - 4 x + 5}{( 2 x ^{2} + 1 ) ^{2}} : \frac{- 10 x ^{2} - 4 x + 5}{2 5 x + 1 ( 2 x ^{2} + 1 ) ^{\frac{3}{2}}}

= \frac{- 10 x ^{2} - 4 x + 5}{2 5 x + 1 ( 2 x ^{2} + 1 ) ^{\frac{3}{2}}}

x \to \infty lim (\frac{3 x ^{2} + 2 x - 8}{4 x ^{4} - 5 x + 4})

x \to 0 lim (\frac{1}{x + 1})

\int (\frac{1}{3}) x^{3} - 5 x d x

\int (6 x^{2} - 4 x + 5) d x

\int \frac{1}{x ^{3} + 9 x} d x

What is the derivative of sqrt((5x+1/(2x^2+1))) ?
The derivative of sqrt((5x+1/(2x^2+1))) is (-10x^2-4x+5)/(2sqrt(5x+1)(2x^2+1)^{3/2)}
What is the first derivative of sqrt((5x+1/(2x^2+1))) ?
The first derivative of sqrt((5x+1/(2x^2+1))) is (-10x^2-4x+5)/(2sqrt(5x+1)(2x^2+1)^{3/2)}