What is the derivative of the square root of x?

For x > 0, (√(x))' = 1/(2√(x)). For a composite argument, (√(u))' = (u')/(2√(u)).

What is the derivative of tan x?

(tan x)' = 1/(cos^2 x) = 1 + tan^2 x, defined wherever cos x ≠ 0. Likewise, (cot x)' = -1/(sin^2 x).

What is the quotient rule formula?

If u and v are functions with v ≠ 0, then (u/v)' = (u'v - uv')/(v^2). Keep the numerator in the order u'v - uv' — swapping the terms flips the sign.

Derivative Rules and Formulas: The Complete Table

Q: How do you differentiate a composite function?

Use the chain rule: if y = f(u) and u = u(x), then y'_x = f'(u) · u'_x. In words, differentiate the outer function with respect to u, then multiply by the derivative of the inner function with respect to x.

Derivatives are the core of first-year calculus and show up everywhere: analyzing functions, finding maxima and minima, and writing equations of tangent lines. This article collects a complete table of derivative formulas — from basic power and root functions to trigonometric, exponential, and logarithmic functions — along with the four differentiation rules and the chain rule, all with detailed worked examples.

What is a derivative?

The derivative of a function $y = f (x)$ at a point $x_{0}$ is the limit (when it exists):

$f^{'} (x_{0}) = Δ x \to 0 lim \frac{f ( x _{0} + Δ x ) - f ( x _{0} )}{Δ x}$

Geometrically, $f^{'} (x_{0})$ is the slope of the tangent line to the graph at $x_{0}$ . In practice you never compute this limit by hand every time — instead you apply the formulas in the tables below.

Table of basic derivatives

These are the core formulas to memorize. The right-hand column gives the composite (chain rule) form for when the argument is some function $u = u (x)$ rather than just $x$ .

Function	Derivative with respect to $x$	Composite form (with $u$ )
$C$ (constant)	$0$	—
$x$	$1$	—
$x^{n}$	$n x^{n - 1}$	$(u^{n})^{'} = n u^{n - 1} \cdot u^{'}$
$\frac{1}{x}$	$- \frac{1}{x ^{2}}$	$(\frac{1}{u})^{'} = - \frac{u ^{'}}{u ^{2}}$
$x$	$\frac{1}{2 x}$	$(u)^{'} = \frac{u ^{'}}{2 u}$

Note that the power rule for $x^{n}$ works for any real exponent, including negative and fractional ones. For example, $x = x^{1/2}$ , so $(x)^{'} = \frac{1}{2} x^{- 1/2} = \frac{1}{2 x}$ .

Derivatives of trigonometric functions

Function	Derivative with respect to $x$	Composite form
$sin x$	$cos x$	$(sin u)^{'} = u^{'} cos u$
$cos x$	$- sin x$	$(cos u)^{'} = - u^{'} sin u$
$tan x$	$\frac{1}{cos ^{2} x}$	$(tan u)^{'} = \frac{u ^{'}}{cos ^{2} u}$
$cot x$	$- \frac{1}{sin ^{2} x}$	$(cot u)^{'} = - \frac{u ^{'}}{sin ^{2} u}$

Watch the signs: the derivatives of both $cos x$ and $cot x$ carry a minus sign. You can also write $(tan x)^{'} = 1 + tan^{2} x$ .

Derivatives of exponential and logarithmic functions

Function	Derivative with respect to $x$	Composite form
$e^{x}$	$e^{x}$	$(e^{u})^{'} = u^{'} e^{u}$
$a^{x}$	$a^{x} ln a$	$(a^{u})^{'} = u^{'} a^{u} ln a$
$ln x$	$\frac{1}{x}$	$(ln u)^{'} = \frac{u ^{'}}{u}$
$lo g_{a} x$	$\frac{1}{x ln a}$	$(lo g_{a} u)^{'} = \frac{u ^{'}}{u ln a}$

For exponential and logarithmic functions the base must satisfy $a > 0$ and $a \neq = 1$ ; also, $ln x$ and $lo g_{a} x$ are only defined for $x > 0$ .

The four differentiation rules

Starting from the basic formulas, you combine functions using these four rules (with $u = u (x)$ , $v = v (x)$ , and $k$ a constant):

Rule	Formula
Sum / difference	$(u \pm v)^{'} = u^{'} \pm v^{'}$
Constant multiple	$(k u)^{'} = k \cdot u^{'}$
Product	$(uv)^{'} = u^{'} v + u v^{'}$
Quotient	$(\frac{u}{v})^{'} = \frac{u ^{'} v - u v ^{'}}{v ^{2}}$ with $v \neq = 0$

In the quotient rule the numerator is $u^{'} v - u v^{'}$ — the order matters, and swapping the terms flips the sign.

The chain rule for composite functions

If $y = f (u)$ where $u = u (x)$ , then:

$y_{x}^{'} = f^{'} (u) \cdot u_{x}^{'}$

The idea: differentiate the outer function with respect to $u$ , then multiply by the derivative of the inner function with respect to $x$ . This is the key to handling any complicated expression such as $sin (3 x)$ , $(2 x + 1)^{5}$ , or $e^{x^{2}}$ .

Worked examples

Example 1 — a polynomial. Differentiate $y = 3 x^{4} - 5 x^{2} + 2 x - 7$ .

Apply the sum/difference rule and the power rule term by term:

$y^{'} = 3 \cdot 4 x^{3} - 5 \cdot 2 x + 2 = 12 x^{3} - 10 x + 2$

Example 2 — the quotient rule. Differentiate $y = \frac{2 x + 1}{x - 3}$ .

Let $u = 2 x + 1$ (so $u^{'} = 2$ ) and $v = x - 3$ (so $v^{'} = 1$ ):

$y^{'} = \frac{u ^{'} v - u v ^{'}}{v ^{2}} = \frac{2 ( x - 3 ) - ( 2 x + 1 ) \cdot 1}{( x - 3 ) ^{2}} = \frac{2 x - 6 - 2 x - 1}{( x - 3 ) ^{2}} = \frac{- 7}{( x - 3 ) ^{2}}$

Example 3 — the chain rule. Differentiate $y = (x^{2} + 1)^{5}$ .

The outer function is $u^{5}$ and the inner function is $u = x^{2} + 1$ with $u^{'} = 2 x$ :

$y^{'} = 5 u^{4} \cdot u^{'} = 5 (x^{2} + 1)^{4} \cdot 2 x = 10 x (x^{2} + 1)^{4}$

Example 4 — trig combined with the chain rule. Differentiate $y = sin (3 x)$ .

With $u = 3 x$ and $u^{'} = 3$ :

$y^{'} = u^{'} cos u = 3 cos (3 x)$

Example 5 — a composite exponential. Differentiate $y = e^{x^{2}}$ .

With $u = x^{2}$ and $u^{'} = 2 x$ :

$y^{'} = u^{'} e^{u} = 2 x e^{x^{2}}$

Common mistakes

Forgetting to multiply by the inner derivative. Writing $(sin 3 x)^{'} = cos 3 x$ is wrong; the correct answer is $3 cos 3 x$ because you must also multiply by $u^{'} = 3$ .
Getting the sign wrong for $cos x$ and $cot x$ . Both pick up a minus sign when differentiated.
Swapping the order in the quotient rule. The numerator must be $u^{'} v - u v^{'}$ , not $u v^{'} - u^{'} v$ .
Treating the product rule as a product of derivatives. $(uv)^{'} \neq = u^{'} v^{'}$ ; the correct formula is $u^{'} v + u v^{'}$ .
Dropping the $ln a$ for bases other than $e$ . $(a^{x})^{'} = a^{x} ln a$ , not just $a^{x}$ .

Tips to remember

For any composite function, always ask "what is the inner function?" and then multiply by its derivative — this is the single most common place students lose points.
$e^{x}$ is special: its derivative is itself.
Pair each trig derivative with its sign: $sin \to cos$ (positive), $cos \to sin$ (negative).

Master this table and the four rules and you can differentiate almost any function you'll meet in the course. Type any of the examples above back into the calculator to check each step of your solution.