en

English

Español

Português

Français

Deutsch

Italiano

Русский

中文(简体)

한국어

日本語

Tiếng Việt

עברית

العربية

Popular >

derivative of (x^{ln(x)})

Solution

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

Solution

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

Solution steps

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

Apply exponent rule:

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

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

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

Apply the chain rule:

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

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

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

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

Simplify

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

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

Popular Examples

integral of arctan(y)

\int arctan (y) d y

tangent of f(x)=x^4+5e^x,(0,5)tangent of

f (x) = x^{4} + 5 e^{x}, (0, 5)

cos^2(x)sin(x)((dy)/(dx))+(cos^3(x))y=1

cos^{2} (x) sin (x) (\frac{d y}{d x}) + (cos^{3} (x)) y = 1

limit as x approaches 0 of x^n*cos(3/x)

x \to 0 lim (x^{n} \cdot cos (\frac{3}{x}))

integral of (sqrt(4-9x))/x

\int \frac{4 - 9 x}{x} d x

Frequently Asked Questions (FAQ)

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