derivative of (3x+1ln(x-1))

解答

\frac{d}{d x} ((3 x + 1) ln (x - 1))

3 ln (x - 1) + \frac{3 x + 1}{x - 1}

求解步骤

\frac{d}{d x} ((3 x + 1) ln (x - 1))

使用微分乘法定则:

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

f = 3 x + 1, g = ln (x - 1)

= \frac{d}{d x} (3 x + 1) ln (x - 1) + \frac{d}{d x} (ln (x - 1)) (3 x + 1)

\frac{d}{d x} (3 x + 1) = 3

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

= 3 ln (x - 1) + \frac{1}{x - 1} (3 x + 1)

\frac{1}{x - 1} (3 x + 1) = \frac{3 x + 1}{x - 1}

= 3 ln (x - 1) + \frac{3 x + 1}{x - 1}