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

The derivative of x^{ln(3x}) is 3^{ln(x)}x^{ln(x)-1}ln(3x^2)

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

The first derivative of x^{ln(3x}) is 3^{ln(x)}x^{ln(x)-1}ln(3x^2)

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

3^{l n (x)} x^{l n (x) - 1} ln (3 x^{2})

Solution steps

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

Apply exponent rule:

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

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

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

Apply the chain rule:

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

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

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

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

Simplify

e^{l n (3 x) l n (x)} \frac{ln ( 3 x ^{2} )}{x} : 3^{l n (x)} x^{l n (x) - 1} ln (3 x^{2})

= 3^{l n (x)} x^{l n (x) - 1} ln (3 x^{2})

What is the derivative of x^{ln(3x}) ?
The derivative of x^{ln(3x}) is 3^{ln(x)}x^{ln(x)-1}ln(3x^2)
What is the first derivative of x^{ln(3x}) ?
The first derivative of x^{ln(3x}) is 3^{ln(x)}x^{ln(x)-1}ln(3x^2)