Rules for Derivatives of Functions on Open Subsets of ℝⁿ

In the setting of Multivariable Calculus: Notation, let $n$, $m$ and $U$ be as in Multivariable Calculus: Notation (ii), and let $a\in U$. ($Df(a)$ and $\nabla f(a)$ of functions differentiable at $a$ are as in The Derivative Is the Jacobian Matrix, and Is Unique (ii) and The Derivative Is the Jacobian Matrix, and Is Unique (i); sums, scalar multiples and products of functions are as in Pointwise Operations on Real-Valued Maps (i), formed in the same way for $\mathbb{R}^{m}$-valued functions.)

(i) Constant functions. For $c\in\mathbb{R}^{m}$, the constant function $f:U\to\mathbb{R}^{m}$, $f(x)=c$, is differentiable at $a$ with $Df(a)=0_{m\times n}$. For $c\in\mathbb{R}$, the constant function $\varphi:U\to\mathbb{R}$, $\varphi(x)=c$, is differentiable at $a$ with $\nabla\varphi(a)=0$.

(ii) Affine functions. For $B\in\mathcal{M}_{m\times n}(\mathbb{R})$ and $c\in\mathbb{R}^{m}$, the function $f:U\to\mathbb{R}^{m}$, $f(x)=Bx+c$, is differentiable at $a$ with

$$Df(a)=B.$$

For $v\in\mathbb{R}^{n}$ and $c\in\mathbb{R}$, the function $\varphi:U\to\mathbb{R}$, $\varphi(x)=v\cdot x+c$, is differentiable at $a$ with

$$\nabla\varphi(a)=v.$$

(iii) Coordinate functions. For $i\in[n]$, the function $\pi_{i}:U\to\mathbb{R}$, $\pi_{i}(x)=x_{i}$, is differentiable at $a$ with $\nabla\pi_{i}(a)=e_{i}$.

(iv) Sums and scalar multiples. Let $f,g:U\to\mathbb{R}^{m}$ be differentiable at $a$, and let $\lambda\in\mathbb{R}$. Then $f+g$ and $\lambda f$ are differentiable at $a$, and

$$D(f+g)(a)=Df(a)+Dg(a),\qquad D(\lambda f)(a)=\lambda\,Df(a).$$

(v) Sums and scalar multiples, real-valued. Let $f,g:U\to\mathbb{R}$ be differentiable at $a$, and let $\lambda\in\mathbb{R}$. Then $f+g$ and $\lambda f$ are differentiable at $a$, and

$$\nabla(f+g)(a)=\nabla f(a)+\nabla g(a),\qquad \nabla(\lambda f)(a)=\lambda\,\nabla f(a).$$

(vi) Products. Let $f,g:U\to\mathbb{R}$ be differentiable at $a$. Then $fg$ is differentiable at $a$, and

$$\nabla(fg)(a)=f(a)\,\nabla g(a)+g(a)\,\nabla f(a).$$

(vii) Reciprocals. Let $g:U\to\mathbb{R}$ be differentiable at $a$, with $g(x)\neq0$ for every $x\in U$. Then $1/g:U\to\mathbb{R}$, $x\mapsto1/g(x)$, is differentiable at $a$, and

$$\nabla\Bigl(\frac{1}{g}\Bigr)(a)=-\frac{1}{g(a)^{2}}\,\nabla g(a).$$

(viii) Quotients. Let $f,g:U\to\mathbb{R}$ be differentiable at $a$, with $g(x)\neq0$ for every $x\in U$. Then $f/g:U\to\mathbb{R}$, $x\mapsto f(x)/g(x)$, is differentiable at $a$, and

$$\nabla\Bigl(\frac{f}{g}\Bigr)(a)=\frac{1}{g(a)^{2}}\bigl(g(a)\,\nabla f(a)-f(a)\,\nabla g(a)\bigr).$$