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Question Number 57997 by peter frank last updated on 15/Apr/19

$$findallsecondorderpartial$$$$derivatives$$$$f(x,y)={xe}^{x^y}.y^x$$

Commented by MJS last updated on 16/Apr/19

$$\mathrm{for}x:$$$$f=uvw$$$$f^{\prime }=u^{\prime }vw+{uv}^{\prime }w+{uvw}^{\prime }$$$$f^{″}=u^{″}vw+{uv}^{″}w+{uvw}^{″}+2(u^{\prime }{v}^{\prime }w+u^{\prime }{vw}^{\prime }+{uv}^{\prime }{w}^{\prime })$$$$u=x{u}^{\prime }=1u^{″}=0$$$$v=y^x{v}^{\prime }=y^x\ln y{v}^{″}=y^x{\left(\ln y\right)}^2$$$$w={\mathrm{e}}^{x^y}{w}^{\prime }=x^{y-1}y{\mathrm{e}}^{x^y}{w}^{″}=x^{y-2}y((x^y+1)y-1){\mathrm{e}}^{x^y}$$

Commented by MJS last updated on 16/Apr/19

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