Here's the problem:
Let $f : R^n \to R$ be a twice continuously differentiable function. Let $\phi(t) = f(u + td)$ be a composition function from $R$ to $R$, with given vectors $u, d \in R^n$. Express $\phi′(t)$ and $\phi′′(t)$ in terms of the gradient and Hessian of $f$ and the vectors $u, d$.
It's been awhile since I've taken calculus, and I'm unsure if i'm approaching this correctly. Would someone mind checking my work and letting me know what I might be doing wrong?
Well, $\phi$ is just a function of one variable. By chain rule, $$\phi'(t)=\nabla f(u+td)\cdot d.$$ What can you say now about $\phi''$?