If$\mathit{w}\mathbf{=}\mathit{f}\mathbf{(}\mathit{a}\mathit{x}\mathbf{+}\mathit{b}\mathit{y}\mathbf{)}$, show that$\mathit{b}\frac{\mathbf{\partial }\mathbf{w}}{\mathbf{\partial }\mathbf{x}}\mathbf{-}\mathit{a}\frac{\mathbf{\partial }\mathbf{w}}{\mathbf{\partial }\mathbf{y}}\mathbf{=}\mathbf{0}$

Given$w=f(ax+by)$

Partial differentiation is defined as the process, in which find the partial derivative of a function.

In Partial differentiation, the function has more than one variable and find the partial derivative of a function with respect to one variable and keeping the other variable constant.

Find the partial differentiation of function,$w=f(ax+by)$ with respect to x.

$\frac{\partial w}{\partial x}=a{f}^{\text{'}}(ax+by)$ …(1)

Find the partial differentiation of function,$w=f(ax+by)$ with respect to y.

$\frac{\partial w}{\partial y}=b{f}^{\text{'}}(ax+by)$ …(2)

Multiply withb in equation (1) and multiply with in equation (2).

$b\frac{\partial w}{\partial x}=ab{f}^{\text{'}}(ax+by)$ …(3)

$a\frac{\partial w}{\partial y}=ab{f}^{\text{'}}(ax+by)$ …(4)

Now subtract equation (3) from equation (4).

$b\frac{\partial w}{\partial x}-a\frac{\partial w}{\partial y}=0$

Hence it is proved that $b\frac{\partial w}{\partial x}-a\frac{\partial w}{\partial y}=0$