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

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

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

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

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

Solution steps

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

Treat

y

as a constant

Apply exponent rule:

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

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

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

Apply the chain rule:

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

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

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

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

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

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

\frac{\partial}{\partial x} (y^{6} cos (4 x) \cdot 4)

n = 1 \sum \infty \frac{9}{10 0 ^{n}}

x \to 0 lim (g (x))

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

\int e^{7 x} sin (7 x) d x

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