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

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

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

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

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

Solution steps

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

Take the constant out:

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

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

Apply the chain rule:

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

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

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

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

Simplify

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

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

x \to \infty lim (9 x^{2} + x - 3)

\int \frac{1}{y ^{3} + y} d y

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

\int \frac{1}{( x + 1 ) x ^{2} + 2 x - 8} d x

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

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