What is (\partial)/(\partial x)(2/3 (x^2+1)^{3/2}) ?

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

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

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

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

Solution steps

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

Take the constant out:

(a \cdot f)^{'} = a \cdot f^{'}

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

Apply the chain rule:

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

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

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

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

Simplify

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

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

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