y=tan(x+y) 求二阶导数

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y=tan(x+y) 求二阶导数

y'= dy/dx =sec^2(x+y)·(1+y');
→[sec^2(x+y) -1]·y'=sec^2(x+y);
→[tan^2(x+y) ]·y'=sec^2(x+y);
→y'=1/sin^2(x+y);
则:
y'' =dy' /dx
=d[sin^(-2)(x+y)] /dx
=(-2)·sin^(-3)(x+y) ·cos(x+y)·(1+y')
=-2·sin^(-3)(x+y) ·cos(x+y)·[1+sin^(-2)(x+y)]
=-2·cos(x+y)·[sin^(-3)(x+y) +sin^(-5)(x+y)]