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)

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derivative of

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

(\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)

e^{1} - x

\frac{( 2 x + 3 )}{( x ^{3} - 5 )}

4 cos (4 x^{2})

\frac{d}{d t} (\frac{( sin ( t ) )}{( 1 - cos ( t ) )})

e^{3} x

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)