What is the derivative of x^{1-x} ?

The derivative of x^{1-x} is x^{1-x}(-ln(x)+(1-x)/x)

What is the first derivative of x^{1-x} ?

The first derivative of x^{1-x} is x^{1-x}(-ln(x)+(1-x)/x)

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

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

Solution steps

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

Apply exponent rule:

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

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

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

Apply the chain rule:

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

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

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

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

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

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

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