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Question 12.9.3: Find an expression for dz in terms of dx and dy for the foll......

Find an expression for dz in terms of dx and dy for the following functions:
(a)  z = Ax^{a} + By^{b}; (b) z = e^{xu}  \ \text{with}   u = u(x, y);\ \text{ and}    (c) z = ln(x^{2} + y).

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({a})\mathrm{d}z=A\,\mathrm{d}(x^{a})+B\,\mathrm{d}(y^{b})=A a x^{a-1}\,\mathrm{d}x+B b y^{b-1}\,\mathrm{d}y

(b) Arguing directly, using abbreviated notation that drops (x, y) throughout, one has

\mathrm{d}z=e^{x u}\,\mathrm{d}(x u)=e^{x u}(x\mathrm{d}u+u\,\mathrm{d}x)=e^{x u}\{x[u_{1}^{\prime}(x,y)\,\mathrm{d}x+u_{2}^{\prime}(x,y)\,\mathrm{d}y]+u\,\mathrm{d}x\}

 

\quad=e^{x u}\{[x u_{1}^{\prime}(x,y)+u]\,\mathrm{d}x+x u_{2}^{\prime}(x,y)\,\mathrm{d}y\}

 

(\mathrm c)\ dz=d\ln(x^{2}+y)={\frac{\mathrm{d}(x^{2}+y)}{x^{2}+y}}={\frac{2x\,\mathrm{d}x+\mathrm{d}y}{x^{2}+y}}.

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