2. Buktikan bahwa: (a) Bila z=sqrt (x^2+y^2) maka ((partial z)/(partial x))x+((partial z)/(partial y))=z (b) Bila z=lnsqrt (x^2+y^2) maka ((partial z)/(partial x))x+((partial z)/(partial y))=1 (c) Bila z=e^(y)/(x)sin((x)/(y))+c^(2)/(x)cos((y)/(x)) maka (partial z)/(partial x)+(partial z)/(partial y)=0 (d) Bila z=(xy)/(x-y) maka x^2(partial ^2z)/(partial x^2)+2xy(partial ^2z)/(partial xpartial y)=(partial )/(partial x)+y^2(partial ^2z)/(partial y^2)=0 (e) Bila z=e^-4(sinx+cosy) maka (partial ^2z)/(partial x^2)+(partial ^2z)/(partial y^2)+(partial z)/(partial t)

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2. Buktikan bahwa: (a) Bila z=sqrt (x^2+y^2) maka ((partial z)/(partial x))x+((partial z)/(partial y))=z (b) Bila z=lnsqrt (x^2+y^2) maka ((partial z)/(partial x))x+((partial z)/(partial y))=1 (c) Bila z=e^(y)/(x)sin((x)/(y))+c^(2)/(x)cos((y)/(x)) maka (partial z)/(partial x)+(partial z)/(partial y)=0 (d) Bila z=(xy)/(x-y) maka x^2(partial ^2z)/(partial x^2)+2xy(partial ^2z)/(partial xpartial y)=(partial )/(partial x)+y^2(partial ^2z)/(partial y^2)=0 (e) Bila z=e^-4(sinx+cosy) maka (partial ^2z)/(partial x^2)+(partial ^2z)/(partial y^2)+(partial z)/(partial t)

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Mari kita buktikan setiap bagian satu per satu.(a) Bila \(z=\sqrt{x^{2}+y^{2}}\), maka \(\left(\frac{\partial z}{\partial x}\right)x+\left(\frac{\partial z}{\partial y}\right)y=z\).Kita akan mencari turunan parsial dari \(z\) terhadap \(x\) dan \(y\), lalu memeriksa apakah persamaan tersebut benar.1. Mencari \(\frac{\partial z}{\partial x}\): \[z = (x^2 + y1/2}\] Menggunakan aturan rantai, kita dapatkan: \[\frac{\partial z}{\partial x} = \frac{1}{2}(x^2 + y^2)^{-1/2} \cdot 2x = \frac{x}{\sqrt{x^2 + y^2}}\]2. Mencari \(\frac{\partial z}{\partial y}\): \[\frac{\partial z}{\partial y} = \frac{1}{2}(x^2 + y^2)^{-1/2} \cdot 2y = \frac{y}{\sqrt{x^2 + y^2}}\]3. Memeriksa persamaan: \[\left(\frac{\partial z}{\partial x}\right \left(\frac{\partial z}{\partial y}\rightfrac{x}{\sqrt{x^2 + y^2}} \cdot x + \frac{y}{\sqrt{x^2 + y^2}} \cdot y\] \[= \frac{x^2}{\sqrt{x^2 + y^2}} + \frac{y^2}{\sqrt{x^2 + y^2}} = \frac{x^2 + y^2}{\sqrt{x^2 + y^2}} = \sqrt{x^2 + y^2} = z\]Jadi, bagian (a) terbukti benar.(b) Bila \(z=\ln\sqrt{x^{2}+y^{2}}\), maka \(\left(\frac{\partial z}{\partial x}\right)x+\left(\frac{\partial z}{\partial y}\right)y=1\).1. Mencari \(\frac{\partial z}{\partial x}\): \[z = \frac{1}{2}\ln(x^2 + y^2)\] \[\frac{\partial z}{\partial x} = \frac{1}{2(x^2 + y^2)} \cdot 2x = \frac{x}{x^2 + y^2}\]2. Mencari \(\frac{\partial z}{\partial y}\): \[\frac{\partial z}{\partial y} = \frac{1}{2(x^2 + y^2)} \cdot 2y = \frac{y}{x^2 + y^2}\]3. Memeriksa persamaan: \[\left(\frac{\partial z}{\partial x}\right)x + \left(\frac{\partial z}{\partial y}\right)y = \frac{x}{x^2 + y^2} \cdot x + \frac{y}{x^2 + y^2} \cdot y\] \[= \frac{x^2}{x^2 + y^2} + \frac{y^2}{x^2 + y^2} = \frac{x^2 + y^2}{x^2 + y^2} = 1\]Jadi, bagian (b) terbukti benar.(c) Bila \(z=e^{\frac{y}{x}}\sin(\frac{x}{y})+e^{\frac{2}{x}}\cos(\frac{y}{x})\), maka \(\frac{\partial z}{\partial x}+\frac{\partial z}{\partial y}=0\).1. Mencari \(\frac{\partial z}{\partial x}\): \[z = e^{\frac{y}{x}}\sin(\frac{x}{y}) + e^{\frac{2}{x}}\cos(\frac{y}{x})\] \[\frac{\partial z}{\partial x} = e^{\frac{y}{x}}\cos(\frac{x}{y}) \cdot \left(-\frac{y}{x^2}\right) + e^{\frac{2}{x}}\sin(\frac{x}{y}) \cdot \left(-\frac{2}{x^2}\right)\]2. Mencari \(\frac{\partial z}{\partial y}\): \

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