40.G. Let f:R→R be differentiable on R. (a) If F(x,y)=f(xy), then xD1F(x,y)=yD2F(x,y) for all (x,y). (b) If F(x,y)=f(ax+by) where a,b∈R, then bD1F(x,y)=aD2F(x,y) for all (x,y). (c) If F(x,y)=f(x2+y2), then yD1F(x,y)=xD2F(x,y) for all (x,y). (d) If F(x,y)=f(x2−y2), then yD1F(x,y)+xD2F(x,y)=0 for all (x,y).

Let's go through each part step by step. (a) If F(x,y)=f(x.y) , we want to show that xD_1F(x,y)=yD_2F(x,y) for all (x,y) To prove this, we n...

Resuelto 40.G. Let f:R→R be differentiable on R. (a) If

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Pregunta: 40.G. Let f:R→R be differentiable on R. (a) If F(x,y)=f(xy), then xD1F(x,y)=yD2F(x,y) for all (x,y). (b) If F(x,y)=f(ax+by) where a,b∈R, then bD1F(x,y)=aD2F(x,y) for all (x,y). (c) If F(x,y)=f(x2+y2), then yD1F(x,y)=xD2F(x,y) for all (x,y). (d) If F(x,y)=f(x2−y2), then yD1F(x,y)+xD2F(x,y)=0 for all (x,y).

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Paso 1
Let's go through each part step by step.

(a) If $F (x, y) = f (x . y)$ , we want to show that $x D_{1} F (x, y) = y D_{2} F (x, y)$ for all $(x, y)$
Explanation:
To prove this, we n...
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Texto de la transcripción de la imagen:

40.G. Let

f : R \to R

be differentiable on

R

. (a) If

F (x, y) = f (x y)

, then

x D_{1} F (x, y) = y D_{2} F (x, y)

for all

(x, y)

. (b) If

F (x, y) = f (a x + b y)

where

a, b \in R

, then

b D_{1} F (x, y) = a D_{2} F (x, y)

for all

(x, y)

. (c) If

F (x, y) = f (x^{2} + y^{2})

, then

y D_{1} F (x, y) = x D_{2} F (x, y)

for all

(x, y)

. (d) If

F (x, y) = f (x^{2} - y^{2})

, then

y D_{1} F (x, y) + x D_{2} F (x, y) = 0

for all

(x, y)