What is the derivative of log_{3}(x^{sqrt(x)}) ?

The derivative of log_{3}(x^{sqrt(x)}) is (ln(x)+2)/(2ln(3)sqrt(x))

What is the first derivative of log_{3}(x^{sqrt(x)}) ?

The first derivative of log_{3}(x^{sqrt(x)}) is (ln(x)+2)/(2ln(3)sqrt(x))

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\frac{d}{d x} (lo g_{3} (x^{x}))

\frac{ln ( x ) + 2}{2 ln ( 3 ) x}

Solution steps

\frac{d}{d x} (lo g_{3} (x^{x}))

Apply log rule:

lo g_{a} (b) = \frac{ln ( b )}{ln ( a )}

= \frac{d}{d x} \frac{ln ( x ^{x} )}{ln ( 3 )}

Take the constant out:

(a \cdot f)^{'} = a \cdot f^{'}

= \frac{1}{ln ( 3 )} \frac{d}{d x} (ln (x^{x}))

Apply the chain rule:

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

= \frac{1}{x ^{x}} \frac{d}{d x} (x^{x})

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

= \frac{1}{ln ( 3 )} \cdot \frac{1}{x ^{x}} x^{x} (\frac{ln ( x )}{2 x} + \frac{1}{x})

Simplify

\frac{1}{ln ( 3 )} \cdot \frac{1}{x ^{x}} x^{x} (\frac{ln ( x )}{2 x} + \frac{1}{x}) : \frac{ln ( x ) + 2}{2 ln ( 3 ) x}

= \frac{ln ( x ) + 2}{2 ln ( 3 ) x}

\int 20 t^{3} d t

x \to 28.3 lim (f (x))

\frac{8}{3 s ^{2} + 12}

\int tan^{3} (6 x) sec^{10} (6 x) d x

\frac{d ^{2}}{d x ^{2}} (x (2 x + 3)^{5})

What is the derivative of log_{3}(x^{sqrt(x)}) ?
The derivative of log_{3}(x^{sqrt(x)}) is (ln(x)+2)/(2ln(3)sqrt(x))
What is the first derivative of log_{3}(x^{sqrt(x)}) ?
The first derivative of log_{3}(x^{sqrt(x)}) is (ln(x)+2)/(2ln(3)sqrt(x))