How to proof this equation, if $f \in C^1(\mathbb R^2)$?

Question

Proof this equation if f є С¹(ℝ²). It was asked in topic Taylor`s Formula for multivariable functions. Can you help me to proof it? Click for image.

$$f(x,y)=f(0,0)+\int_0^1(f'_x(tx,ty)x+f'_y(tx,ty)y)dt$$

Answer 1

This has nothing to do with Taylor above the fact that both are infinitesimal calculus.

Define $h(t)=f(tx,ty)$. Then by the fundamental theorem of calculus $$ f(x,y)-f(0,0)=h(1)-h(0)=\int_0^1 h'(t) dt $$ For the derivative of $h$ you need then the multidimensional chain rule. This gives the form of the integrand in the claim.