Differentiation Rules and Table of Derivatives
Every derivative – no matter how complicated the function looks – is built from a handful of differentiation rules and a short list of basic derivatives. This page collects both. Each rule links to a detailed explanation with examples, and every example can be opened in the step-by-step derivative calculator.
📚 The differentiation rules
Here $u$ and $v$ are differentiable functions of $x$, $c$ and $n$ are constants.
| Rule | Formula | Use it for |
|---|---|---|
| Constant Rule | $\frac{d}{dx}c = 0$ | The derivative of any constant is zero. |
| Derivative of x | $\frac{d}{dx}x = 1$ | The derivative of the variable itself is one. |
| Constant Factor Rule | $(c\cdot f)' = c\cdot f'$ | Constant factors stay in front of the derivative. |
| Sum and Difference Rule | $(u \pm v)' = u' \pm v'$ | Differentiate sums and differences term by term. |
| Power Rule | $\frac{d}{dx}x^n = n\,x^{n-1}$ | Differentiate powers like $x^n$, roots and $1/x^n$. |
| Product Rule | $(uv)' = u'v + uv'$ | For products of two functions $u\cdot v$. |
| Quotient Rule | $\left(\frac{u}{v}\right)' = \frac{u'v - uv'}{v^2}$ | For fractions of two functions $u/v$. |
| Chain Rule | $\big(f(g(x))\big)' = f'(g(x))\,g'(x)$ | For nested (composite) functions $f(g(x))$. |
📋 Table of derivatives of basic functions
Memorise these – together with the rules above they are enough to differentiate every elementary function.
Powers, roots, exponentials and logarithms
| Function $f(x)$ | Derivative $f'(x)$ | Explained |
|---|---|---|
| $c$ (constant) | $0$ | constant rule |
| $x$ | $1$ | derivative of x |
| $x^n$ | $n\,x^{n-1}$ | power rule |
| $\dfrac{1}{x}$ | $-\dfrac{1}{x^2}$ | power rule |
| $\sqrt{x}$ | $\dfrac{1}{2\sqrt{x}}$ | root rule |
| $\sqrt[n]{x}$ | $\dfrac{1}{n\sqrt[n]{x^{\,n-1}}}$ | root rule |
| $e^x$ | $e^x$ | exponential functions |
| $a^x$ | $a^x \ln a$ | exponential functions |
| $\ln x$ | $\dfrac{1}{x}$ | derivative of ln x |
| $\ln|x|$ | $\dfrac{1}{x}$ | derivative of ln x |
| $\log_a x$ | $\dfrac{1}{x\ln a}$ | logarithm rule |
Trigonometric functions
| Function | Derivative | Function | Derivative |
|---|---|---|---|
| $\sin x$ | $\cos x$ | $\cos x$ | $-\sin x$ |
| $\tan x$ | $\sec^2 x = \dfrac{1}{\cos^2 x}$ | $\cot x$ | $-\csc^2 x = -\dfrac{1}{\sin^2 x}$ |
| $\sec x$ | $\sec x\tan x$ | $\csc x$ | $-\csc x\cot x$ |
Derivatives of trig functions explained, with proofs →
Inverse trigonometric functions
| Function | Derivative | Function | Derivative |
|---|---|---|---|
| $\arcsin x$ | $\dfrac{1}{\sqrt{1-x^2}}$ | $\arccos x$ | $-\dfrac{1}{\sqrt{1-x^2}}$ |
| $\arctan x$ | $\dfrac{1}{1+x^2}$ | $\operatorname{arccot} x$ | $-\dfrac{1}{1+x^2}$ |
| $\operatorname{arcsec} x$ | $\dfrac{1}{|x|\sqrt{x^2-1}}$ | $\operatorname{arccsc} x$ | $-\dfrac{1}{|x|\sqrt{x^2-1}}$ |
Derivatives of inverse trig functions with derivations →
Hyperbolic functions
| Function | Derivative | Function | Derivative |
|---|---|---|---|
| $\sinh x$ | $\cosh x$ | $\cosh x$ | $\sinh x$ |
| $\tanh x$ | $\operatorname{sech}^2 x = 1-\tanh^2 x$ | $\coth x$ | $-\operatorname{csch}^2 x$ |
| $\operatorname{arsinh} x$ | $\dfrac{1}{\sqrt{x^2+1}}$ | $\operatorname{artanh} x$ | $\dfrac{1}{1-x^2}$ |
Derivatives of sinh, cosh and tanh explained →
📈 Proofs and graphs for single functions such as arctan, tan, sec or $x^x$: derivatives of common functions.
🧭 Which rule do I need?
Look at the outermost operation of the function – the one you would evaluate last:
- A sum or difference? Differentiate term by term with the sum rule; constant factors stay (constant factor rule).
- A product of two functions? Use the product rule $(uv)' = u'v + uv'$ – e.g. $x^2\sin x$.
- A fraction with $x$ in the denominator? Use the quotient rule – or rewrite $\frac{1}{x^n}$ as $x^{-n}$ and use the power rule.
- A function inside another function? Use the chain rule: outer derivative times inner derivative – e.g. $\sin(3x)$, $e^{x^2}$, $\ln(x^2+1)$.
- A basic function? Look it up in the table above.
Most exercises combine several rules. The calculator shows exactly in which order they are applied – for example $x^2e^{3x}$ (product + chain rule) or $\ln(\sin x)$ (chain rule twice).
✍️ Notation
The derivative of $y = f(x)$ is written in several equivalent ways:
$f'(x)$ is Lagrange's notation, $\frac{dy}{dx}$ is Leibniz's notation. Leibniz's form makes the chain rule easy to remember: $\frac{dy}{dx} = \frac{dy}{du}\cdot\frac{du}{dx}$. The second derivative is written $f''(x)$ or $\frac{d^2y}{dx^2}$.
❓ Frequently asked questions
What are the basic rules of differentiation?
The basic rules are the constant rule ($c' = 0$), the power rule ($(x^n)' = n x^{n-1}$), the constant factor rule, the sum rule, the product rule, the quotient rule and the chain rule. Together with the derivatives of $e^x$, $\ln x$ and the trigonometric functions they cover every elementary function.
What is the difference between the product rule and the chain rule?
The product rule is used when two functions are multiplied, like $x^2\cdot\sin x$. The chain rule is used when one function is inside another, like $\sin(x^2)$. Many functions need both, e.g. $x\cdot e^{2x}$.
What is the derivative of a constant?
Zero. A constant does not change, so its slope is $0$: $\frac{d}{dx}7 = 0$, $\frac{d}{dx}\pi = 0$.
Do I need the quotient rule for every fraction?
No. If only a constant is in the denominator, use the constant factor rule: $\left(\frac{x^3}{5}\right)' = \frac{3x^2}{5}$. If only a power of $x$ is in the denominator, rewrite it as a negative power: $\frac{2}{x^3} = 2x^{-3}$, so the derivative is $-6x^{-4}$.