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If xyz" = k, show that, at x=y=z,22(Karay =-(x log ex)" 1. |
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Answer» First, recall that (d/dt) t^t = t^t (1 + ln t) via logarithmic differentiation. Let y = t^t.==> ln y = t ln t. Differentiate both sides:(1/y) dy/dt = 1 + ln t==> dy/dt = y(1 + ln t) = t^t (1 + ln t).----------------Remembering that z is a function of x, differentiate both sides with respect to x:(x^x (1 + ln x)) * y^y * z^z + x^x y^y * z^z (1 + ln z) ∂z/∂x = 0==> x^x y^y z^z (1 + ln z) ∂z/∂x = -x^x (1 + ln x) y^y z^z==> (1 + ln z) ∂z/∂x = -(1 + ln x). Similarly, (1 + ln z) ∂z/∂y = -(1 + ln y). Differentiate both sides of (1 + ln z) ∂z/∂y = -(1 + ln y) with respect to x:(1/z) ∂z/∂x * ∂z/∂y + (1 + ln z) * ∂²z/∂x∂y = 0==> (1 + ln z) ∂²z/∂x∂y = (-1/z) [-(1 + ln x)/(1 + ln z)] * [-(1 + ln y)/(1 + ln z)]==> ∂²z/∂x∂y = -(1 + ln x)(1 + ln y)/ [z(1 + ln z)^3]. Letting x = y = z yields∂²z/∂x∂y = -(1 + ln x)(1 + ln x)/ [x(1 + ln x)^3].............= -1/[x(1 + ln x)]. |
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