Help with a property of a convex function

Question

I'm studying linear and nonlinear programming and on my book I bumped into the following statement:

$$\lim_{\alpha \to 0} \displaystyle \frac{f(\textbf{x}+\alpha (\textbf{y}-\textbf{x}))-f(\textbf{x})}{\alpha} = \nabla f(\textbf{x})(\textbf{y}-\textbf{x})$$

Could someone show me, why is the statement above true?...I get confused again with this for some reason :-)

Thank you for any help :-)

Answer 1

Well, the equation is saying nothing more that the directional derivative of a differentiable function is equal to the inner product of the direction and the gradient of that function: Your left hand side is the directional derivative of $f$ at $x$ in direction $(y-x)$ and your right hand side is the inner product of the gradient of $f$ at $x$ with the direction $(y-x)$.