How to compute higher order derivative of multivariable function wrt t?

Question

What is the explicit procedure for differentiating multivariable functions with respect to a scalar? For a very simple example, I have a function $f: \mathbb{R}^d \to \mathbb{R}$ and $\phi(t) = f(x + tv)$, $v \in \mathbb{R}^d$. What are the steps to calculating $\phi'(t), \phi''(t), \phi'''(t)$, etc?

Answer 1

Well, consider the function $s : \mathbb R \to \mathbb R^d$ such that $s(t ) = x + tv$. Then $s’(t) = v$. Since $\phi = f\circ s$, you may apply the multivariable chain rule to find $$\phi’(t) = \sum_{i=1}^d \frac{\partial f}{\partial x_i}(s(t))\ s’(t) = \nabla f(x + tv) \cdot v$$ The subsequent derivatives are calculated in a similar way. For example, you may let $F :\mathbb R^d \to \mathbb R$ such that $F(w) = v \cdot \nabla f (w)$ and write $\phi’ = F\circ s$ in order to apply the multivariable chain rule again. (What are the partial derivatives of $F$?)