What is (\partial)/(\partial x)(x^{sqrt(x)}) ?

The answer to (\partial)/(\partial x)(x^{sqrt(x)}) is x^{sqrt(x)}((ln(x))/(2sqrt(x))+1/(sqrt(x)))

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(\partial)/(\partial x)(x^{sqrt(x)})

\frac{\partial}{\partial x} (x^{x})

x^{x} (\frac{ln ( x )}{2 x} + \frac{1}{x})

Solution steps

\frac{\partial}{\partial x} (x^{x})

Apply exponent rule:

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

x^{x} = e^{x l n (x)}

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

Apply the chain rule:

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

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

\frac{\partial}{\partial x} (x ln (x)) = \frac{ln ( x )}{2 x} + \frac{1}{x}

= e^{x l n (x)} (\frac{ln ( x )}{2 x} + \frac{1}{x})

e^{x l n (x)} = x^{x}

= x^{x} (\frac{ln ( x )}{2 x} + \frac{1}{x})

g (x) = 1 + 2 x

n = 1 \sum \infty (\frac{cos ( π )}{3})^{n}

\int_{0}^{\frac{π}{4}} sin^{3} (x) d x

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

16 y^{^{''}} + 8 y^{^{'}} + y = x + 6

What is (\partial)/(\partial x)(x^{sqrt(x)}) ?
The answer to (\partial)/(\partial x)(x^{sqrt(x)}) is x^{sqrt(x)}((ln(x))/(2sqrt(x))+1/(sqrt(x)))