Question

x-sinxcosx/sinx(tanx)^2

Answer 1

我们可以使用三角函数的基本恒等式和求导的方法来计算这个式子的导数。

首先，将被除数 x-sinxcosx 改写为 xcosx - sin^2 x, 并将分母中的 sin(x)(tan(x))^2 改写为 sin^2(x)/cos^2(x)，则原式可以写成：

[(xcosx - sin^2x) / sin(x)] * [cos^2(x) / sin^2(x)]

将这个式子化简：

= [(x cos(x))/sin(x)] * [cos^2(x)/sin^2(x)] - [sin^2(x)/sin(x)] * [cos^2(x)/sin^2(x)]

= x/(sin(x)cos(x)) - tan(x)

然后对这个式子分别对 x 求导，得到：

d/dx [x/(sin(x)cos(x)) - tan(x)] = [sin(x)cos(x)-x(cos^2(x)-sin^2(x))]/(sin^2(x)cos^2(x))

= [sin(x)cos(x) - xcos(2x)] / [sin^2(x)cos^2(x)]

因此，原式的导数是 [sin(x)cos(x) - xcos(2x)] / [sin^2(x)cos^2(x)]。