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

The derivative of (3x^{ln(3x)}) is (2e^{ln^2(3x)}ln(3x))/x

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

The first derivative of (3x^{ln(3x)}) is (2e^{ln^2(3x)}ln(3x))/x

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\frac{d}{d x} ((3 x)^{l n (3 x)})

\frac{2 e ^{l n^{2} (3 x)} ln ( 3 x )}{x}

Solution steps

\frac{d}{d x} ((3 x)^{l n (3 x)})

Apply exponent rule:

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

(3 x)^{l n (3 x)} = e^{l n (3 x) l n (3 x)}

= \frac{d}{d x} (e^{l n (3 x) l n (3 x)})

Apply the chain rule:

e^{l n (3 x) l n (3 x)} \frac{d}{d x} (ln (3 x) ln (3 x))

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

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

= e^{l n (3 x) l n (3 x)} \frac{2 ln ( 3 x )}{x}

Simplify

e^{l n (3 x) l n (3 x)} \frac{2 ln ( 3 x )}{x} : \frac{2 e ^{l n^{2} (3 x)} ln ( 3 x )}{x}

= \frac{2 e ^{l n^{2} (3 x)} ln ( 3 x )}{x}

\frac{d r}{d t} = - 2 t r, r (0) = 2

\int sin^{- \frac{3}{2}} (x) cos^{3} (x) d x

x \to - 1 lim (2 x^{2} - 5 x + 3)

\int \frac{1}{4 + t} d t

\int x e^{10 x} d x

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