请问为什么y=arcsinx,则siny=x呢?

2024-11-24 04:27:33
推荐回答(5个)
回答1:

y=arcsinx的反函数是:x=siny

为了表述上的习惯性,一般说

它的反函数是:

y=sinx

但是在求导数的时候就不能这样了

应该是这样

y=arcsinx的导数

=1/(siny)'

=1/cosy

=1/根号(1-sin^2y)

=1/根号(1-x^2)

简介

一般来说,设函数y=f(x)(x∈A)的值域是C,若找得到一个函数g(y)在每一处g(y)都等于x,这样的函数x= g(y)(y∈C)叫做函数y=f(x)(x∈A)的反函数,记作x=f-1(y) 。反函数x=f-1(y)的定义域、值域分别是函数y=f(x)的值域、定义域。最具有代表性的反函数就是对数函数与指数函数。

一般地,如果x与y关于某种对应关系f(x)相对应,y=f(x),则y=f(x)的反函数为x=f-1(y)。存在反函数(默认为单值函数)的条件是原函数必须是一一对应的(不一定是整个数域内的)。注意:上标"−1"指的是函数幂,但不是指数幂。

回答2:

反函数的导数等于直接函数导数的倒数.(这句话是对的)
y=arcsinx的反函数是:
x=siny
为了表述上的习惯性,一般说
它的反函数是:
y=sinx
但是在求导数的时候就不能这样了
应该是这样
y=arcsinx的导数
=1/(siny)'
=1/cosy
=1/根号(1-sin^2y)
=1/根号(1-x^2)

回答3:

没有为什么,sin是三角函数,arcsin是反三角函数。sinx=y,那么x=arcsiny,相当于2的平方=4,那么2=4的开方。

回答4:

两边同时用正弦,所以siny=sin(arcsinx)=x

回答5:

arcsin是sin的反函数,由反函数定义可知:y=f(x)的反函数是x=f(y)
所以y=arcsinx,那么根据反函数定义,x=siny。

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