x^y-y^x=2,求dy/dx

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x^y-y^x=2,求dy/dx
x^y-y^x=2,求dy/dx

x^y-y^x=2,求dy/dx
令x^y＝u,则：ylnx＝lnu,∴lnxdy/dx＋y/x＝（1/u）du/dx,∴u（lnxdy/dx＋y/x）＝du/dx,
∴du/dx＝x^y·lnx·dy/dx＋（y/x）x^y.
令y^x＝t,则：xlny＝lnt,∴lny＋（x/y）dy/dx＝（1/t）dt/dx,∴t［lny＋（x/y）dy/dx］＝dt/dx,
∴dt/dx＝y^x·lny＋（x/y）y^x·dy/dx.
∵x^y＋y^x＝2,∴u＋t＝2,∴du/dx＋dt/dx＝0,
∴［x^y·lnx·dy/dx＋（y/x）x^y］＋［y^x·lny＋（x/y）y^x·dy/dx］＝0,
∴［x^y·lnx＋（x/y）y^x］dy/dx＝－［（y/x）x^y＋y^x·lny］,
∴dy/dx＝－［（y/x）x^y＋y^x·lny］/［x^y·lnx＋（x/y）y^x］.

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