What is the derivative of \sqrt[4]{(3x-2/(x^2+1)}) ?

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

What is the first derivative of \sqrt[4]{(3x-2/(x^2+1)}) ?

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

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

Solution steps

Apply the chain rule:

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

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

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

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

Simplify

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

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

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