derivative of log_{10}(log_{10}(x))

解答

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

\frac{1}{ln ^{2} ( 10 ) x lo g _{10} ( x )}

求解步骤

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

使用对数计算法则:

lo g_{a} (b) = \frac{ln ( b )}{ln ( a )}

= \frac{d}{d x} (\frac{ln ( lo g _{10} ( x ) )}{ln ( 10 )})

将常数提出:

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

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

使用连锁律:

\frac{1}{lo g _{10} ( x )} \frac{d}{d x} (lo g_{10} (x))

= \frac{1}{lo g _{10} ( x )} \frac{d}{d x} (lo g_{10} (x))

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

= \frac{1}{ln ( 10 )} \cdot \frac{1}{lo g _{10} ( x )} \cdot \frac{1}{x ln ( 10 )}

化简

\frac{1}{ln ( 10 )} \cdot \frac{1}{lo g _{10} ( x )} \cdot \frac{1}{x ln ( 10 )} : \frac{1}{ln ^{2} ( 10 ) x lo g _{10} ( x )}

= \frac{1}{ln ^{2} ( 10 ) x lo g _{10} ( x )}

\int_{0}^{\frac{π}{2}} sin (2 x) d x

l a pl a ce t r an s f or m cos (2 t)

t an g e n t y = \frac{( x ^{2} - 1 )}{( x ^{2} + x + 1 )}

x^{2} y^{^{''}} - x y^{^{'}} + y = 0, y = x

\frac{d}{d x} (e^{x} (tan (x) - x))