求y=5^Intanx与y=x^5x的微分

Question

求y=5^Intanx 与 y=x^5x的微分

Answer 1

y = 5^(lntanx)
lny = [ln(tanx)]×ln5
两边取微分
d(lny) = d[(lntanx)ln5]
(1/y)dy = ln5d(lntanx)
dy = yln5(1/tanx)d(tanx) = (yln5/tanx)(1+(tanx)^2)dx
dy = [y(1+(tanx)^2)ln5/tanx]dx
y=x^5x
lny = 5xlnx
(1/y)dy = (5lnx)dx + 5x(1/x)dx = (5lnx + 5)dx = 5(lnx + 1)dx
dy = 5y(lnx + 1)dx