已知dydx求d2ydx2

鱼栋贱3884设x=5(t−sint)y=5(1−sint),求dydx,d2ydx2. -
慎炉宇15979797290 ______[答案] dy dt= d[5(1−sint)] dt=5(−cost)=−5cost, dx dt= d[5(t−sint)] dt=5(1−cost); ① dy dx= dydt dxdt= −5cost 5(1−cost)=1− 1 1−cost; ② d2y dx2= d(dydx) dx= d(dydx)dt dxdt= d(1−11−cost)dt 5(1−cost)= sint(1−cost)2 5(1−cost)= sint 5(1−cost)3.

鱼栋贱3884设 x=cos(t2)y=tcos(t2)−∫t2112ucosudu,求dydx、d2ydx2在t=π2的值. -
慎炉宇15979797290 ______[答案] ∵dxdt=−2tsin(t2),dydt=cos(t2)−2t2sin(t2)−cos(t2)2t2•2t=-2t2sin(t2),(t>0)∴dydx=dydtdxdt=t∴d2ydx2=ddx(dydx)=ddt(dydx)•dtdx=ddt(dydx)dxdt=1−2tsin(t2)∴d2ydx2|t=π2=−12π...

鱼栋贱3884x∧2y∧3+cosy=0,求dy╱dx -
慎炉宇15979797290 ______ x^2.y^3 +cosy =0 x^2. (3y^2) dy/dx + 2xy^3 - siny .dy/dx =0(3x^2.y^2- siny )dy/dx = -2xy^3 dy/dx =-2xy^3/(3x^2.y^2- siny )

鱼栋贱3884设函数y=y(x)由方程xef(y)=ey确定,其中f具有二阶导数,且f′≠1,求d2ydx2. -
慎炉宇15979797290 ______[答案] ∵xef(y)=ey ∴两边去自然对数得:lnx+f(y)=y 对x求导得 1 x+f′(y)y′=y′ ∴y′= 1 x[1−f′(y)] ∴y″=− 1 x2[1−f′(y)]2•[1−f′(y)−xf″(y)y′] 将y'代入得 y″=− (1−f′(y))2−f″(y) x2[1−f′(y)]3

鱼栋贱3884求解微分方程dydx=2xy. -
慎炉宇15979797290 ______[答案] 由微分方程 dy dx=2xy,得 dy y=2xdx(y≠0) 两边积分得:ln|y|=x2+C1 即y=Cex2(C为任意常数)

鱼栋贱3884设函数y=y(x)由参数方程x=1+2t2y=∫1+2lnt1euudu(t>1)所确定,求d2ydx2|x=9. -
慎炉宇15979797290 ______[答案] 由参数方程x=1+2t2y=∫1+2lnt1euudu∴dxdt=4t,dydt=e1+2lnt1+2lnt•2t=2et1+2lnt∴dydx=dydtdxdt=2et1+2lnt4t=e2(1+2lnt)所以 d2ydx2=ddx(dydx)=ddt(dydx)•dtdx=ddt(dydx)dxdt=−e2(1+2lnt)2•2t•...

鱼栋贱3884设x=∫t0f(u2)duy=[f(t2)]2,其中f(u)具有二阶导数,且f(u)≠0,求d2ydx2. -
慎炉宇15979797290 ______[答案] ∵ x=∫t0f(u2)duy=[f(t2)]2 ∴ dx dt=f(t2), dy dt=2f(t2)•f′(t2)•2t=4tf(t2)f′(t2) ∴ dy dx= dydt dxdt= 4tf(t2)f′(t2) f(t2)=4tf′(t2) ∴ d2y dx2= d dx( dy dx)= d dt( dy dx)• dt dx= ddt(dydx) dxdt═4[f′(t2)+tf″(t2)2t]• 1 f(t2)= 4[f′(t2)+2t2f″(t2)] f(t2)

鱼栋贱3884设x2+2xy - y2=2x,求dydx. -
慎炉宇15979797290 ______[答案] ∵x2+2xy-y2=2x, ∴2x+2y+2xy'-2yy'=2 y'(x-y)=1-x-y, ∴ dy dx= 1-x-y x-y.

鱼栋贱3884设参数方程{ x=arctant y=t - ln(1+t^2),则dy/dx= -
慎炉宇15979797290 ______[答案] dy/dx=(dy/dt)/(dx/dt) dy/dt=1-[2t/(1+t^2)] dx/dt=1/(1+t^2) 故dy/dx=(t-1)^2

鱼栋贱3884设函数y=y(x)由方程y - xey=1所确定,求d2ydx2|x=0的值设函数y=y(x)由方程y - xey=1所确定,求d2ydx2|x=0的值. -
慎炉宇15979797290 ______[答案] 解; 设F(x,y)=y-xey-1,则 Fx=?ey,Fy=1?xey ∴ dy dx=? Fx Fy= ey 1?xey ∴ d2y dx2= d dx( ey 1?xey)= eydydx(1?xey)+ey(ey+xeydydx) (1?xey)2…① 又当x=0时,y=1 ∴ dy dx|x=0=1 将 dy dx|x=0=1代入到①得: d2y dx2|x=0=e(e+1)