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y=ArCtAn(x²)求 y的导数

先求arctanx的导数 y=arctanx,则x=tany arctanx′=1/tany′ tany′=(siny/cosy)′=cosycosy-siny(-siny)/cos²y=1/cos²y 则arctanx′=cos²y=cos²y/sin²y+cos²y=1/(1+tan²y)=1/(1+x²) 所以arctan2x的导数=2ar...

arctanX的导数是1/(1+X²) 这里的X=x/2 复合函数求导,需要先求子函数的导数,即X'=1/2 再乘上arctanX的导数 所以所求导数是1/[2(1+x²/4)]

如图

∂z/∂x= {1/[1+(y/x)²]}(y/x)`= {1/[1+(y/x)²]}(-y/x²) (这是复合函数求导,即要对(y/x)中的x求导, 即(y/x)`=-y/x²)

导数计算如下: y=arctanf(x) y'=f'(x)/[1+f^2(x)]. 复合函数的求导。

解: dy/dx=(dy/dt)/(dx/dt) =[ln(1+t²)]'/(t-arctanx)' =[2t/(1+t²)]/[1- 1/(1+t²)] =2t/(1+t²-1) =2t/t² =2/t d²y/dx²=[d(2/t)/dt]/(dx/dt) =(-2/t²)/[1- 1/(1+t²)] =(-2/t²)/[(1+t²-...

y=arctan{(1-x²)/(1+x²)} = 1 / {1+ (1-x²)²/(1+x²)²} * {(1-x²)/(1+x²)} ′ = 1 / {1+ (1-x²)²/(1+x²)²} * { - 1 + 2/(1+x²)} ′ = 1 / {1+ (1-x²)²/(1+x²)...

先要知道arctanx的导数

y'= (x²+1)'/[1+(x²+1)²]=2x/[1+(x²+1)²]

∂z/∂x=1/(1+y²/x²)*(-y/x²)=-y/(x²+y²) ∂z/∂y=1/(1+y²/x²)*1/x=x/(x²+y²) ∂²z/∂x²=y/(x²+y²)*2x=2xy/(x²+y²)² ∂&...

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