tanX)✀=1⼀(cosX)^2=(secX)^2 是怎么得出的 要有详细的推理过程

d/dx sinx = cosx
d/dx cosx = -sinx

d/dx tanx = d/dx sinx/cosx
= (cosx d/dx sinx - sinx d/dx cosx)/cos²x，导数商法则
= [cosx (cosx) - sinx (-sinx)]/cos²x
= (cos²x + sin²x)/cos²x
= 1/cos²x
= sec²x

定义：
d/dx tanx
= lim(h->0) [f(x + h) - f(x)]/h
= lim(h->0) [tan(x + h) - tanx]/h
= lim(h->0) (1/h)[(tanx + tanh)/(1 - tanx tanh) - tanx]
= lim(h->0) (1/h)[(tanx + tanh - tanx + tan²x tanh)/(1 - tanx tanh)]
= lim(h->0) (1/h)[(tanh + tan²x tanh)/(1 - tanx tanh)]
= lim(h->0) (1/h)tanh(1 + tan²x)/(1 - tanx tanh)
= (1 + tan²x)lim(h->0) sinh/h * 1/cosh * 1/(1 - tanx tanh)
= sec²x * 1 * 1 * 1/(1 - 0)
= sec²x