求y=arcsin(1-2x)的求导过程(详细的) 求函数的导数y=arcsin(1-2x)
\u51fd\u6570\u6c42\u5bfc\uff0cy=arcsin\uff081-2x\uff09\uff0c\u8be6\u7ec6\u6b65\u9aa4 \uff1ay'=1/\u221a[1-(1-2x)²]\u8fd9\u662f\u4e2a\u516c\u5f0f\uff0c\u53ef\u4ee5\u76f4\u63a5\u7528
\u51fd\u6570\u7684\u5bfc\u6570\u7b49\u4e8e\u53cd\u51fd\u6570\u5bfc\u6570\u7684\u5012\u6570\uff0cy=arcsinx,\u5219x=siny\uff0c\u6c42\u5bfc\u4e3acosy\uff0c\u800c\uff0ccosy\u5e73\u65b9+siny\u5e73\u65b9=1\uff0c\u4e8e\u662fcosy=\u6839\u53f7(1-siny\u5e73\u65b9)\uff0c\u5373\u6839\u53f7(1-x^2)\uff0c\u6240\u4ee5y=arcsinx\u6c42\u5bfc\u540e\u4e3a1/\u6839\u53f7(1-x^2)
\u590d\u5408\u51fd\u6570\u6c42\u5bfc\u89c4\u5219\uff0c\u5229\u7528\u94fe\u5f0f\u6cd5\u5219\u6c42\uff0c\u8fd0\u7528\u5e42\u51fd\u6570\uff1a
y\uff1dx\uff3en\uff0cy\uff07\uff1dnx\uff3e\uff08n\uff0d1\uff09
y\uff1darcsinxy\uff07\uff1d1\uff0f\u221a1\uff0dx\uff3e2
y\uff07\uff1d\uff08arcsin\uff081\uff0d2x\uff09\uff09\uff07
\uff1d1\uff0f\u221a1\uff0d\uff081\uff0d2x\uff09\uff3e2
\uff1d1\uff0f2\u221a\uff08x\uff0dx\uff3e2\uff09
\u6216\u8005
y\uff07\uff1d1\uff0f\u221a\uff3b1\uff0d\uff081\uff0d2x\uff09²\uff3d\u00b7\uff081\uff0d2x\uff09\uff07
\uff1d\uff0d2\uff0f\u221a\uff084x\uff0d4x²\uff09
\uff1d\uff0d1\uff0f\u221a\uff08x\uff0dx²\uff09\u3002
\u6269\u5c55\u8d44\u6599
\u5229\u7528\u5bfc\u6570\u5b9a\u4e49\u6c42\u51fd\u6570\u5bfc\u6570\u7684\u65b9\u6cd5\uff1a
\u4f7f\u7528\u5bfc\u6570\u5b9a\u4e49\u6c42\u89e3\u5bfc\u6570\u7684\u6b65\u9aa4\u4e3b\u8981\u5206\u4e3a\u4e09\u4e2a\u6b65\u9aa4\u3002\u8fd9\u91cc\u4ee5\u5e42\u51fd\u6570y\uff1dx\uff3en\u4e3a\u4f8b\u8bf4\u660e\u3002
\u7b2c\u4e00\u6b65\uff0c\u6c42\u51fa\u56e0\u53d8\u91cf\u7684\u589e\u91cf\u0394y\uff1df\uff08x\uff0b\u0394\uff09\uff0df\uff08x\uff09\u3002
\u7b2c\u4e8c\u6b65\uff0c\u8ba1\u7b97\u0394y\u4e0e\u0394x\u7684\u6bd4\u503c\u3002
\u7b2c\u4e09\u6b65\uff0c\u6c42\u6781\u9650\uff0c\u4ee4\u0394x\u8d8b\u8fd1\u4e8e0\uff0c\u53ef\u4ee5\u6c42\u5f97\u6781\u9650\u3002
\u5e42\u51fd\u6570\u7684\u6c42\u89e3\u6bd4\u8f83\u7b80\u5355\u3002\u5bf9\u4e8e\u4e00\u4e9b\u5176\u4ed6\u8f83\u590d\u6742\u7684\u51fd\u6570\uff0c\u8fd8\u9700\u8981\u501f=\u501f\u52a9\u4e00\u4e9b\u6570\u5b66\u516c\u5f0f\u4ee5\u53ca\u6781\u9650\u8fd0\u7b97\u3002\u4f8b\u5982\u5bf9\u4e8ey=sin\uff08x\uff09\u7684\u6c42\u89e3\uff0c\u5c31\u9700\u8981\u5229\u7528\u548c\u5dee\u5316\u79ef\u516c\u5f0f\u4e0elim(x->0){sin\uff08x\uff09/x}=1\u8fd9\u4e24\u4e2a\u516c\u5f0f\u3002
\u540c\u6837\uff0c\u9996\u5148\u8ba1\u7b97\u589e\u91cf\u0394y\uff1df\uff08x\uff0b\u0394\uff09\uff0df\uff08x\uff09\u3002
\u63a5\u4e0b\u6765\u7684\u4e24\u6b65\u53ef\u4ee5\u4e00\u540c\u8fdb\u884c\u3002
具体回答如下：
y'=1/√[1-(1-2x)²] ·(1-2x)'
=-2/√(4x-4x²)
=-1/√(x-x²)
求导的意义：
求导是微积分的基础，同时也是微积分计算的一个重要的支柱。物理学、几何学、经济学等学科中的一些重要概念都可以用导数来表示。
如导数可以表示运动物体的瞬时速度和加速度、可以表示曲线在一点的斜率、还可以表示经济学中的边际和弹性。
siny=1-2x
两边对x求导：cosyy'=-2
所以y'=-2/cosy=-2√（1-siny^2）=-2/√【1-（1-2x）^2】
解：
y'=[arcsin(1-2x)]'
={1/√[1-(1-2x)²]}·(1-2x)'
=-2/√(1-1+4x-4x²)
=-2/[2√(x-x²)]
=-1/√(x-x²)
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绛旓細dy/dx =1/鈭(1-(鈭(1-x^2)^2)) * (-x)/鈭(1-x^2) =1/x * (-x)/鈭(1-x^2) =-x/x * 鈭(1-x^2)
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绛旓細arcsin(1-x)=y 1-x=siny x=1-siny 鈭磃(x)=arcsin(1-x)鐨勫弽鍑芥暟涓y=1-sinx,(-蟺/2鈮も墹蟺/2)
