Question

求助大神，帮我看看这几道高数题怎么做

(打钩的那几道)

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

Answer 1

解如下图所示

Answer 2

3. 求下列各函数的导数：
(2). y=x^a+a^x+x^x=x^a+a^x+e^(xlnx);
y'=ax^(a-1)+(a^x)lna+e^(xlnx)(xlnx)'=ax^(a-1)+(a^x)lna+(x^x)(lnx+1);
(4). (sinx)^y=(cosy)^x；

解一：两边取对数得：yln(sinx)=xln(cosy)；
两边对x取导数得：y'ln(sinx)+y(cosx)/(sinx)=ln(cosy)-xy'(siny)/(cosy);
[ln(sinx)+xtany]y'=ln(cosy)-ycotx； ∴y'=[ln(cosy)-ycotx]/[ln(sinx)+xtany];
解二：F(x，y)=yln(sinx)-xln(cosy)=0
∴y'=-(∂F/∂x)/(∂F/∂y)=-[ycotx-ln(cosy)]/[ln(sinx)+xtany]
=[ln(cosy)-ycotx]/[ln(sinx)+xtany];
4。求d²y/dx²
(1). e^y+xy=e²;
解：(e^y)y'+y+xy'=0；∴y'=-y/(x+e^y)；
y''=[-(x+e^y)y'+y(1+y'e^y)]/(x+e^y)²
=[(-x-e^y+ye^y)y'+y]/(x+e^y)²(将上面已求出的y'=-y/(x+e^y)代入并化简）
=[(xy+ye^y-y²e^y)/(x+e^y)+y]/(x+e^y)²
=(xy+ye^y-y²e^y+xy+ye^y)/(x+e^y)³
=(2xy+2ye^y-y²e^y)/(x+e^y)³;
(2). x=t²-2t；y=t³-3t；求d²y/dx²;
dy/dx=y'=(dy/dt)/(dx/dt)=(3t²-3)/(2t-2)；
d²y/dx²=y''=dy'/dx=(dy'/dt)/(dx/dt)={[6t(2t-2)-2(3t²-3)]/(2t-2)²}/(2t-2)
=(6t²-12t+6)/(2t-2)³=(3t²-6t+3)/[4(t-1)³];