Differentiation Rules

These are the standard rules for computing derivatives of functions $f, g : \mathbb{R} \to \mathbb{R}$ . They are the basic toolkit used whenever partial derivatives are computed component-by-component.

Core Rules

Let $f$ and $g$ be differentiable functions and $c \in \mathbb{R}$ a constant. The fundamental differentiation rules are:

$\frac{d}{dx}[c] = 0 \qquad \frac{d}{dx}[x^n] = n x^{n-1}$

$\frac{d}{dx}[c \cdot f] = c \cdot f' \qquad \frac{d}{dx}[f + g] = f' + g'$

$\frac{d}{dx}[f \cdot g] = f' g + f g' \qquad \frac{d}{dx}\!\left[\frac{f}{g}\right] = \frac{f' g - f g'}{g^2}$

$\frac{d}{dx}[f(g(x))] = f'(g(x)) \cdot g'(x)$

The last rule is the chain rule — it is the most important rule for computing partial derivatives of composite functions.

Common Derivatives

$f(x)$	$f'(x)$
$c$	$0$
$x^n$	$n x^{n-1}$
$e^x$	$e^x$
$e^{ax}$	$a e^{ax}$
$\ln x$	$\dfrac{1}{x}$
$\sin x$	$\cos x$
$\cos x$	$-\sin x$
$\tan x$	$\dfrac{1}{\cos^2 x}$

Chain Rule in Detail

The chain rule handles composite functions. If $h(x) = f(g(x))$ , then:

$h'(x) = f'(g(x)) \cdot g'(x)$

Let $h(x) = e^{-x} \sin x$ . Using the product rule:

$h'(x) = (e^{-x})' \sin x + e^{-x} (\sin x)' = -e^{-x} \sin x + e^{-x} \cos x = e^{-x}(\cos x - \sin x)$

Let $h(x) = \sin(x^2)$ . Using the chain rule with $f(u) = \sin u$ and $g(x) = x^2$ :

$h'(x) = \cos(x^2) \cdot 2x = 2x \cos(x^2)$

Partial Derivatives via These Rules

When computing partial derivatives, all the rules above apply — simply treat every variable other than the one being differentiated as a constant. For example, with $f(x, y, z) = e^{-z} \sin x + y^2$ :

$\frac{\partial f}{\partial x}$ : treat $y, z$ as constants → $e^{-z} \cos x$
$\frac{\partial f}{\partial y}$ : treat $x, z$ as constants → $2y$
$\frac{\partial f}{\partial z}$ : treat $x, y$ as constants, apply chain rule on $e^{-z}$ → $-e^{-z} \sin x$