grandes-ecoles 2014 QIB1

grandes-ecoles · France · centrale-maths1__pc Chain Rule Chain Rule with Composition of Explicit Functions

Let $g$ denote a function of class $C^1$ from $\mathbb{R}^n$ to $\mathbb{R}$. We fix an element $a = (a_1, a_2, \ldots, a_n)$ of $\mathbb{R}^n$. Let $\varphi$ be the function from $\mathbb{R}$ to $\mathbb{R}$ defined by $$\varphi(t) = g(ta) = g(ta_1, ta_2, \ldots, ta_n)$$ Justify that $\varphi$ is of class $C^1$ on $\mathbb{R}$ and, for every real $t$, give $\varphi'(t)$.

Let $g$ denote a function of class $C^1$ from $\mathbb{R}^n$ to $\mathbb{R}$. We fix an element $a = (a_1, a_2, \ldots, a_n)$ of $\mathbb{R}^n$. Let $\varphi$ be the function from $\mathbb{R}$ to $\mathbb{R}$ defined by
$$\varphi(t) = g(ta) = g(ta_1, ta_2, \ldots, ta_n)$$
Justify that $\varphi$ is of class $C^1$ on $\mathbb{R}$ and, for every real $t$, give $\varphi'(t)$.

Paper Questions

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