What is (\partial)/(\partial x)((2x)^x) ?

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

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(\partial)/(\partial x)((2x)^x)

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

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

Solution steps

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

Apply exponent rule:

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

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

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

Apply the chain rule:

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

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

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

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

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

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

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