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

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

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

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

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

Solution steps

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

Treat

n, y, z

as constants

Apply the chain rule:

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

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

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

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

Simplify

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

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

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

\frac{\partial}{\partial x} (2 1 - y^{2})

(14 + x) \frac{d y}{d x} = 5 y

\frac{d y}{d x} = \frac{( x - 1 )}{y - 1}

\frac{d v}{d t} - \frac{4 v}{t} = 8 t^{3}

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