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

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

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

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

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

Solution steps

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

Treat

y, z

as constants

Apply the chain rule:

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

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

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

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

Simplify

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

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

\frac{d}{d x} (x^{8 e^{5 x}})

\frac{d}{d x} (12 x (x^{2} + 10)^{5})

\int 2 sin^{2} (x) cos (x) d x

\int 2 x x + 3 d x

\int x^{e} + e^{x} d x

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