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)

English

Español

Português

Français

Deutsch

Italiano

Русский

中文(简体)

한국어

日本語

Tiếng Việt

עברית

العربية

(\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})

\int (5 - 3 x)^{10} d x

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

im pl i c i t e^{\frac{x}{y}} = 5 x - y

l a pl a ce t r an s f or m (t - 2)^{2}

\int_{1}^{4} \frac{x ^{2} + 8}{5 x - x ^{2}} 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)