What is the derivative of (5sqrt(x)^{ln(x)}) ?

The derivative of (5sqrt(x)^{ln(x)}) is (x^{(ln(25x)-2)/2}ln(25x^2))/(2)

What is the first derivative of (5sqrt(x)^{ln(x)}) ?

The first derivative of (5sqrt(x)^{ln(x)}) is (x^{(ln(25x)-2)/2}ln(25x^2))/(2)

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\frac{d}{d x} ((5 x)^{l n (x)})

\frac{x ^{\frac{l n ( 25 x ) - 2}{2}} ln ( 25 x ^{2} )}{2}

Solution steps

\frac{d}{d x} ((5 x)^{l n (x)})

Apply exponent rule:

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

(5 x)^{l n (x)} = e^{l n (x) l n (5 x)}

= \frac{d}{d x} (e^{l n (x) l n (5 x)})

Apply the chain rule:

e^{l n (x) l n (5 x)} \frac{d}{d x} (ln (x) ln (5 x))

= e^{l n (x) l n (5 x)} \frac{d}{d x} (ln (x) ln (5 x))

\frac{d}{d x} (ln (x) ln (5 x)) = \frac{ln ( 5 x )}{x} + \frac{ln ( x )}{2 x}

= e^{l n (x) l n (5 x)} (\frac{ln ( 5 x )}{x} + \frac{ln ( x )}{2 x})

Simplify

e^{l n (x) l n (5 x)} (\frac{ln ( 5 x )}{x} + \frac{ln ( x )}{2 x}) : \frac{x ^{\frac{l n ( 25 x ) - 2}{2}} ln ( 25 x ^{2} )}{2}

= \frac{x ^{\frac{l n ( 25 x ) - 2}{2}} ln ( 25 x ^{2} )}{2}

\int sin (2 x) cos (2 x) d x

f (x) = \frac{1}{5} x - \frac{1}{2}

\int 8 x^{7} d x

\frac{x ^{\frac{3}{2}}}{3}

\frac{e ^{- 2 s}}{s ( 3 s + 2 )}

What is the derivative of (5sqrt(x)^{ln(x)}) ?
The derivative of (5sqrt(x)^{ln(x)}) is (x^{(ln(25x)-2)/2}ln(25x^2))/(2)
What is the first derivative of (5sqrt(x)^{ln(x)}) ?
The first derivative of (5sqrt(x)^{ln(x)}) is (x^{(ln(25x)-2)/2}ln(25x^2))/(2)