Partial Derivatives Analysis?
Suppose f: ℝ → ℝ is differentiable on ℝ and define a fnc g: ℝ^2 → ℝ by
g(x,y) = f(x+y).
Show that ∂g/∂x = ∂g/∂y on ℝ^2
Thanks
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Suppose f: ℝ → ℝ is differentiable on ℝ and define a fnc g: ℝ^2 → ℝ by
g(x,y) = f(x+y).
Show that ∂g/∂x = ∂g/∂y on ℝ^2
Thanks
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Verified answer
Let u(x,y) = x + y
∂u/∂x = ∂u/∂y = 1
g(x,y) = f(u(x,y))
∂g/∂x = ∂f/∂u ∂u/∂x = ∂f/∂u
∂g/∂y = ∂f/∂u ∂u/∂y = ∂f/∂u
==> ∂g/∂x = ∂g/∂y