F(x+y)=f(x)f(y)

Question

Find partial differentiation

Answer 1

F(x+y)=f(x)f(y)

∂F/∂x=f(y)df/dx, ∂F/∂y=f(x)df/dy.

f(y) is a constant when partially differentiating with respect to x.

f(x) is a constant when partially differentiating with respect to y.

EXAMPLE

Let f(x)=3x²+1 so f(y)=3y²+1, df/dx=6x, df/dy=6y.

F(x+y)=(3x²+1)(3y²+1)=9x²y²+3x²+3y²+1,

∂F/∂x=18xy²+6x=6x(3y²+1), ∂F/∂y=18x²y+6y=6y(3x²+1).

These partial differentials confirm the general case since 6x=df/dx, 6y=df/dy, 3y²+1=f(y), 3x²+1=f(x).