Find the first order partial derivatives \frac{\partial w}{\partial x}\ and\ \frac{\partial...

Question:

Find the first order partial derivatives {eq}\frac{\partial w}{\partial x}\ and\ \frac{\partial w}{\partial y} {/eq} for the following implicit relation: {eq}w^3 - xy + yw + y^3 - 2= 0 {/eq} at (1,1,1).

Partial Derivative

The derivative of the composition of the two or more variables functions with respect to one taking other as the constant is known as the partial derivative.

Answer and Explanation:

{eq}\Rightarrow \ w^{3}-xy+yw+y^{3}-2=0\\ \text{differentiate with respect to x taking other as contant to get }\frac{\delta{w}}{\delta{x}}\\ \Rightarrow \ \frac{\delta{w}}{\delta{x}}=\frac{\delta}{\delta{x}}(w^{3}-xy+yw+y^{3}-2)\\ \Rightarrow \ \frac{\delta{w}}{\delta{x}}=-y\\ \text{value of }\frac{\delta{w}}{\delta{x}} \ at \ (1,1,1)\\ \Rightarrow \ \frac{\delta{w}}{\delta{x}}=-1\\ \Rightarrow \ w^{3}-xy+yw+y^{3}-2=0\\ \text{differentiate with respect to y taking other as contant to get }\frac{\delta{w}}{\delta{y}}\\ \Rightarrow \ \frac{\delta{w}}{\delta{y}}=\frac{\delta}{\delta{y}}(w^{3}-xy+yw+y^{3}-2)\\ \Rightarrow \ \frac{\delta{w}}{\delta{y}}=-x+w+3y^{2}\\ \text{value of }\frac{\delta{w}}{\delta{y}} \ at \ (1,1,1)\\ \Rightarrow \ \frac{\delta{w}}{\delta{y}}=-1+1+3\\ \Rightarrow \ \frac{\delta{w}}{\delta{y}}=3\\ {/eq}


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Double Integrals & Evaluation by Iterated Integrals

from GRE Math: Study Guide & Test Prep

Chapter 15 / Lesson 4
401

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