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{d}{d x} (\frac{( - 2 x ^{2} + 2 x )}{( 3 x - 3 )})

\frac{d}{d x} (\frac{40}{3 x ^{\frac{1}{3}}})

x \to 0 lim (\frac{x + 4 - 2}{sin ( 5 x )})

\int 39 arctan (x) d x

\int 6 t sin (t) d t

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)