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?

dannyboi2006tk 2022-09-29 Answered
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 + t v ), v R d . What are the steps to calculating ϕ ( t ) , ϕ ( t ) , ϕ ( t ), etc?
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Answers (1)

Allvin03
Answered 2022-09-30 Author has 7 answers
Well, consider the function s : R R d such that s ( t ) = v. Then 𝑠′(𝑡)=𝑣. Since ϕ = f s, you may apply the multivariable chain rule to find
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