y=arcsinX、arccosX、arctanX、arccotX的导数.如题

Question

y=arcsinX、arccosX、arctanX、arccotX的导数.
如题

Answer 1

都换成反函数,再用复合函数求导法.
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y = arcsinx
siny = x
cosy * y' = 1
y' = 1/cosy = 1/√(1 - sin²y) = 1/√(1 - x²)
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y = arccosx
cosy = x
- siny * y' = 1
y' = - 1/siny = - 1/√(1 - cos²y) = - 1/√(1 - x²)
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y = arctanx
tany = x
sec²y * y' = 1
y' = 1/sec²y = 1/(1 + tan²y) = 1/(1 + x²)
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y = arccotx
coty = x
- csc²y * y' = 1
y' = - 1/csc²y = - 1/(1 + cot²y) = - 1/(1 + x²)