derivative 7xln((16x)/(16x+1))

解答

导数

7 x ln (\frac{16 x}{16 x + 1})

7 (ln (\frac{16 x}{16 x + 1}) + \frac{1}{16 x + 1})

求解步骤

\frac{d}{d x} (7 x ln (\frac{16 x}{16 x + 1}))

将常数提出:

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

= 7 \frac{d}{d x} (x ln (\frac{16 x}{16 x + 1}))

使用微分乘法定则:

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

f = x, g = ln (\frac{16 x}{16 x + 1})

= 7 (\frac{d x}{d x} ln (\frac{16 x}{16 x + 1}) + \frac{d}{d x} (ln (\frac{16 x}{16 x + 1})) x)

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

\frac{d}{d x} (ln (\frac{16 x}{16 x + 1})) = \frac{1}{x ( 16 x + 1 )}

= 7 (1 \cdot ln (\frac{16 x}{16 x + 1}) + \frac{1}{x ( 16 x + 1 )} x)

化简

7 (1 \cdot ln (\frac{16 x}{16 x + 1}) + \frac{1}{x ( 16 x + 1 )} x) : 7 (ln (\frac{16 x}{16 x + 1}) + \frac{1}{16 x + 1})

= 7 (ln (\frac{16 x}{16 x + 1}) + \frac{1}{16 x + 1})

\frac{d y}{d x} = 2 x + 3 y

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