derivative of xln(1-x)

解答

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

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

求解步骤

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

使用微分乘法定则:

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

f = x, g = ln (1 - x)

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

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

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

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

化简

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

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

n \to \infty lim (2^{n})

\frac{\partial}{\partial x} (e^{3 x})

x \to - 9 + lim (\frac{6 x + 54}{x + 9})

a re a 9 x^{4} - 9 x^{2}, 10 x^{2}

\frac{d}{d x} ((\frac{1}{x ^{2}} - \frac{7}{x ^{4}}) (x + 3 x^{3}))