Differentiation

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:

Rule	Formula
Constant	$\frac{d}{dx}[c] = 0$
Power	$\frac{d}{dx}[x^n] = n x^{n-1}$
Constant multiple	$\frac{d}{dx}[c \cdot f] = c \cdot f'$
Sum	$\frac{d}{dx}[f + g] = f' + g'$
Product	$\frac{d}{dx}[f \cdot g] = f' g + f g'$
Quotient	$\frac{d}{dx}\!\left[\frac{f}{g}\right] = \frac{f' g - f g'}{g^2}$
Chain	$\frac{d}{dx}[f(g(x))] = f'(g(x)) \cdot g'(x)$

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

Common Derivatives

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

Trigonometric Derivatives

The trigonometric derivatives follow from the rules above combined with the limit definitions of $\sin$ and $\cos$ . The full set:

The derivatives of the trigonometric functions are:

$f(x)$	$f'(x)$
$\sin x$	$\cos x$
$\cos x$	$-\sin x$
$\tan x$	$\dfrac{1}{\cos^2 x} = \sec^2 x$
$\cot x$	$-\dfrac{1}{\sin^2 x} = -\csc^2 x$
$\sec x$	$\sec x \tan x$
$\csc x$	$-\csc x \cot x$

For the inverse trigonometric functions:

\frac{d}{dx} \arcsin x = \frac{1}{\sqrt{1 - x^2}} \qquad \frac{d}{dx} \arccos x = -\frac{1}{\sqrt{1 - x^2}} \qquad \frac{d}{dx} \arctan x = \frac{1}{1 + x^2}

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)

Translation Invariance

For any fixed constant $a \in \mathbb{R}$ , the shifted variable $(x - a)$ behaves under differentiation exactly like $x$ alone — a property called translation invariance:

\frac{d}{dx}(x - a) = 1

This follows from the constant rule and sum rule together: $\frac{d}{dx}[x - a] = \frac{d}{dx}[x] - \frac{d}{dx}[a] = 1 - 0 = 1$ . As a consequence, all the standard rules apply to expressions in $(x - a)$ exactly as they apply to expressions in $x$ :

\frac{d}{dx}[(x - a)^n] = n(x - a)^{n-1} \qquad \frac{d}{dx}[f(x - a)] = f'(x - a)

The same holds in higher dimensions: for fixed $\mathbf{a} \in \mathbb{R}^n$ , the Jacobian of the shift is the identity matrix,

\frac{\partial}{\partial \mathbf{x}}(\mathbf{x} - \mathbf{a}) = I

so derivatives of expressions in $(\mathbf{x} - \mathbf{a})$ behave exactly like derivatives of expressions in $\mathbf{x}$ alone.

This is what makes Taylor expansions around any base point — in powers of $(x - a)$ or $(\mathbf{x} - \mathbf{a})$ — no harder to manipulate than expansions around the origin.

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 $e^{-z}$ → $-e^{-z} \sin x$