Calculate the derivative of $f$ in the direction of a unitary vector.

Question

Let $T:\mathbb{R}^2\rightarrow\mathbb{R}^2$ a linear transformation. Consider scalar field in$\mathbb{R}^2$ defined by $f(\vec{x})=<\vec{x},T\vec{x}>$. Calculate the derivative of $f$ in the direction of a unitary vector.

I don't have a clear idea of how to attack this type of exercise. Can someone give me a hint of how to solve this exercise?

Answer 1

HINT

Let consider and expand

$$f(\vec{x_0} + \vec h)=<\vec{x_0}+\vec h,T(\vec{x_0}+\vec h)>$$

to show that

$$\nabla f(\vec x_0)=(T+T^T)\cdot \vec x_0$$