# What is the explicit procedure for differentiating multivariable functions with respect to a scalar? For a very simple example, I have a function f:R^d->R and ϕ(t)=f(x+tv), v in R^d. What are the steps to calculating ϕ′(t),ϕ″(t),ϕ‴(t), etc?

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 $\varphi \left(t\right)=f\left(x+tv\right)$, $v\in {\mathbb{R}}^{d}$. What are the steps to calculating ${\varphi }^{\prime }\left(t\right),{\varphi }^{″}\left(t\right),{\varphi }^{‴}\left(t\right)$, etc?
You can still ask an expert for help

• Questions are typically answered in as fast as 30 minutes

Solve your problem for the price of one coffee

• Math expert for every subject
• Pay only if we can solve it

Allvin03
Well, consider the function $s:\mathbb{R}\to {\mathbb{R}}^{d}$ such that ${s}^{\prime }\left(t\right)=v$. Then 𝑠′(𝑡)=𝑣. Since $\varphi =f\circ s$, you may apply the multivariable chain rule to find