What is the derivative of xln(1/x) ?

The derivative of xln(1/x) is -ln(x)-1

What is the first derivative of xln(1/x) ?

The first derivative of xln(1/x) is -ln(x)-1

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

- ln (x) - 1

Solution steps

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

Simplify

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

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

Take the constant out:

(a \cdot f)^{'} = a \cdot f^{'}

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

Apply the Product Rule:

(f \cdot g)^{'} = f^{'} \cdot g + f \cdot g^{'}

f = x, g = ln (x)

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

\frac{d x}{d x} = 1

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

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

Simplify

- (1 \cdot ln (x) + \frac{1}{x} x) : - ln (x) - 1

= - ln (x) - 1

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