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

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

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

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

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

Solution steps

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

Treat

y, z

as constants

Apply exponent rule:

a^{b} = e^{b l n (a)}

x^{x^{2} + y^{2} + z^{2}} = e^{(x^{2} + y^{2} + z^{2}) l n (x)}

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

Apply the chain rule:

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

= e^{(x^{2} + y^{2} + z^{2}) l n (x)} \frac{\partial}{\partial x} ((x^{2} + y^{2} + z^{2}) ln (x))

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

= e^{(x^{2} + y^{2} + z^{2}) l n (x)} (2 x ln (x) + \frac{x ^{2} + y ^{2} + z ^{2}}{x})

e^{(x^{2} + y^{2} + z^{2}) l n (x)} = x^{x^{2} + y^{2} + z^{2}}

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

sin (2 x) d x + cos (6 y) d y = 0, y (\frac{π}{2}) = \frac{π}{6}

\int e^{3 x + 5} d x

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

\int \frac{1}{x ^{- 0.1}} d x

x \to \infty lim (x \cdot (arctan (e^{x}) - \frac{π}{2}))

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