What is (\partial)/(\partial x)(sqrt(x^3+x)) ?

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

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(\partial)/(\partial x)(sqrt(x^3+x))

\frac{\partial}{\partial x} (x^{3} + x)

\frac{3 x ^{2} + 1}{2 x ^{3} + x}

Solution steps

\frac{\partial}{\partial x} (x^{3} + x)

Apply the chain rule:

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

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

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

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

Simplify

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

= \frac{3 x ^{2} + 1}{2 x ^{3} + x}

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