derivative y=log_{10}(sin^2(x))

解答

导数

y = lo g_{10} (sin^{2} (x))

\frac{sin ( 2 x )}{ln ( 10 ) sin ^{2} ( x )}

求解步骤

\frac{d}{d x} (lo g_{10} (sin^{2} (x)))

使用对数计算法则:

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

= \frac{d}{d x} (\frac{ln ( sin ^{2} ( x ) )}{ln ( 10 )})

将常数提出:

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

= \frac{1}{ln ( 10 )} \frac{d}{d x} (ln (sin^{2} (x)))

使用连锁律:

\frac{1}{sin ^{2} ( x )} \frac{d}{d x} (sin^{2} (x))

= \frac{1}{sin ^{2} ( x )} \frac{d}{d x} (sin^{2} (x))

\frac{d}{d x} (sin^{2} (x)) = sin (2 x)

= \frac{1}{ln ( 10 )} \cdot \frac{1}{sin ^{2} ( x )} sin (2 x)

化简

\frac{1}{ln ( 10 )} \cdot \frac{1}{sin ^{2} ( x )} sin (2 x) : \frac{sin ( 2 x )}{ln ( 10 ) sin ^{2} ( x )}

= \frac{sin ( 2 x )}{ln ( 10 ) sin ^{2} ( x )}

d er i v a t i v e \frac{sin ^{2} ( x )}{x}

\frac{d}{d t} (3 t^{3} - 2 t^{2} + t - 5)

d er i v a t i v e \frac{x ^{4}}{4 + x ^{33} + x ^{22}}

d er i v a t i v e f (x) = x^{3} + 1

d er i v a t i v e \frac{8 x}{3}