derivative of 10xlog_{10}(100x)

解答

\frac{d}{d x} (10 x lo g_{10} (100 x))

10 (lo g_{10} (x) + 2 + \frac{1}{ln ( 10 )})

求解步骤

\frac{d}{d x} (10 x lo g_{10} (100 x))

化简

10 x lo g_{10} (100 x) : 10 x (lo g_{10} (x) + 2)

= \frac{d}{d x} (10 x (lo g_{10} (x) + 2))

将常数提出:

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

= 10 \frac{d}{d x} (x (lo g_{10} (x) + 2))

使用微分乘法定则:

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

f = x, g = lo g_{10} (x) + 2

= 10 (\frac{d x}{d x} (lo g_{10} (x) + 2) + \frac{d}{d x} (lo g_{10} (x) + 2) x)

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

\frac{d}{d x} (lo g_{10} (x) + 2) = \frac{1}{ln ( 10 ) x}

= 10 (1 \cdot (lo g_{10} (x) + 2) + \frac{1}{ln ( 10 ) x} x)

化简

10 (1 \cdot (lo g_{10} (x) + 2) + \frac{1}{ln ( 10 ) x} x) : 10 (lo g_{10} (x) + 2 + \frac{1}{ln ( 10 )})

= 10 (lo g_{10} (x) + 2 + \frac{1}{ln ( 10 )})

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