What is the derivative of (x^x) ?

The derivative of (x^x) is x^x(ln(x)+1)

What is the first derivative of (x^x) ?

The first derivative of (x^x) is x^x(ln(x)+1)

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

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

Solution steps

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

Apply exponent rule:

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

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

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

Apply the chain rule:

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

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

\frac{d}{d x} (x ln (x)) = ln (x) + 1

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

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

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

y^{^{'}} + \frac{y}{10} = 500

\frac{d N}{d t} = N (1 - 0.0002 N)

\frac{d}{d x} (c_{1} cos (2 x) + c_{2} sin (2 x))

\frac{d}{d x} (sec (\frac{1}{2} x) tan (\frac{1}{2} x))

\frac{d}{d x} (\frac{a}{x ^{2} + b})

What is the derivative of (x^x) ?
The derivative of (x^x) is x^x(ln(x)+1)
What is the first derivative of (x^x) ?
The first derivative of (x^x) is x^x(ln(x)+1)