91Ó°ÊÓ

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.

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

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

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

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

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

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

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

$b\frac{∂w}{∂x}-a\frac{∂w}{∂y}=0$

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