[Math] Finding directional derivative given directional derivative in another direction

Question

I know how to find the directional derivative given a function but how do I find the derivative given a directional derivative is another direction?

Answer 1

Given a function $f(x)$ and its gradient $g=\nabla f,\,$ we can calculate the derivatives of $f$ in the direction of two distinct unit vectors $(a,b)$ $$\eqalign{ \alpha &= g^Ta,\quad\beta &= g^Tb,\qquad \{a,b,g\}\in{\mathbb R}^{2} }$$ These vectors form a basis for ${\mathbb R}^{2}$ so any other unit vector can be written as a linear combination. $$\eqalign{ c &= \lambda a + \mu b \\ }$$ Multiplying by our two basis vectors yields a linear system which can be solved for $(\lambda,\mu)$ $$ \begin{array}{rr} \lambda&+&(a^Tb)\,\mu &= &a^Tc\\ (b^Ta)\,\lambda&+&\mu &= &b^Tc \\ \end{array} \\ $$ while multiplying by the gradient, yields the answer to your question: $$\eqalign{ \gamma &= g^Tc \;=\; \lambda\alpha + \mu\beta \\ }$$