What is the derivative of (4x^{ln(4x)}) ?

The derivative of (4x^{ln(4x)}) is (2e^{ln^2(4x)}ln(4x))/x

What is the first derivative of (4x^{ln(4x)}) ?

The first derivative of (4x^{ln(4x)}) is (2e^{ln^2(4x)}ln(4x))/x

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\frac{d}{d x} ((4 x)^{l n (4 x)})

\frac{2 e ^{l n^{2} (4 x)} ln ( 4 x )}{x}

Solution steps

\frac{d}{d x} ((4 x)^{l n (4 x)})

Apply exponent rule:

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

(4 x)^{l n (4 x)} = e^{l n (4 x) l n (4 x)}

= \frac{d}{d x} (e^{l n (4 x) l n (4 x)})

Apply the chain rule:

e^{l n (4 x) l n (4 x)} \frac{d}{d x} (ln (4 x) ln (4 x))

= e^{l n (4 x) l n (4 x)} \frac{d}{d x} (ln (4 x) ln (4 x))

\frac{d}{d x} (ln (4 x) ln (4 x)) = \frac{2 ln ( 4 x )}{x}

= e^{l n (4 x) l n (4 x)} \frac{2 ln ( 4 x )}{x}

Simplify

e^{l n (4 x) l n (4 x)} \frac{2 ln ( 4 x )}{x} : \frac{2 e ^{l n^{2} (4 x)} ln ( 4 x )}{x}

= \frac{2 e ^{l n^{2} (4 x)} ln ( 4 x )}{x}

What is the derivative of (4x^{ln(4x)}) ?
The derivative of (4x^{ln(4x)}) is (2e^{ln^2(4x)}ln(4x))/x
What is the first derivative of (4x^{ln(4x)}) ?
The first derivative of (4x^{ln(4x)}) is (2e^{ln^2(4x)}ln(4x))/x