en

English

Español

Português

Français

Deutsch

Italiano

Русский

中文(简体)

한국어

日本語

Tiếng Việt

עברית

العربية

Popular >

derivative of (1/x ^x)

Solution

\frac{d}{d x} ((\frac{1}{x})^{x})

Solution

(\frac{1}{x})^{x} (- ln (x) - 1)

Solution steps

\frac{d}{d x} ((\frac{1}{x})^{x})

Apply exponent rule:

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

(\frac{1}{x})^{x} = e^{x l n (\frac{1}{x})}

= \frac{d}{d x} (e^{x l n (\frac{1}{x})})

Apply the chain rule:

e^{x l n (\frac{1}{x})} \frac{d}{d x} (x ln (\frac{1}{x}))

= e^{x l n (\frac{1}{x})} \frac{d}{d x} (x ln (\frac{1}{x}))

\frac{d}{d x} (x ln (\frac{1}{x})) = - ln (x) - 1

= e^{x l n (\frac{1}{x})} (- ln (x) - 1)

e^{x l n (\frac{1}{x})} = (\frac{1}{x})^{x}

= (\frac{1}{x})^{x} (- ln (x) - 1)

Graph

Popular Examples

slope of x^2+3 slope

x^{2} + 3

derivative of f(x)=(120x^2)/(x^2+4)derivative of

f (x) = \frac{120 x ^{2}}{x ^{2} + 4}

limit as x approaches pi/2-of tan(x)

x \to \frac{π}{2} - lim (tan (x))

derivative of sqrt(7x^3-7x^2-(9/x))derivative of

7 x^{3} - 7 x^{2} - (\frac{9}{x})

(\partial)/(\partial x)(-xy(3-x-y))

\frac{\partial}{\partial x} (- x y (3 - x - y))

Frequently Asked Questions (FAQ)

What is the derivative of (1/x ^x) ?
The derivative of (1/x ^x) is (1/x)^x(-ln(x)-1)
What is the first derivative of (1/x ^x) ?
The first derivative of (1/x ^x) is (1/x)^x(-ln(x)-1)