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Popular >

derivative of x^{x-1}

Solution

derivative of

x^{x - 1}

Solution

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

Solution steps

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

Apply exponent rule:

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

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

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

Apply the chain rule:

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

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

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

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

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

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

Popular Examples

(\partial)/(\partial x)(xln(z^6+x^4))

\frac{\partial}{\partial x} (x ln (z^{6} + x^{4}))

derivative of y=(6-xe^x)/(x+e^x)derivative of

y = \frac{6 - x e ^{x}}{x + e ^{x}}

laplacetransform 2*e^{(5*t-2)}laplace transform

2 \cdot e^{(5 \cdot t - 2)}

integral from 1/7 to 5 of 10xln(7x)

\int_{\frac{1}{7}}^{5} 10 x ln (7 x) d x

(\partial)/(\partial x)(sin(y/(x+y)))

\frac{\partial}{\partial x} (sin (\frac{y}{x + y}))

Frequently Asked Questions (FAQ)

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