# 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.

{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}