🇷🇴 RO | 🇬🇧 EN
Derivatives
Graphs
Inverse Trig
Hyperbolic
Derivative Rules
Higher Derivatives
Taylor Series
Integrals

Derivative Table & Domains

Common functions, domains of definition and differentiability conditions.
\(f(x)\)\(D_f\)Conditions\(f'(x)\)
\(C\)\(\mathbb{R}\)—\(0\)
\(x^n\)\(\mathbb{R}\)\(n\in\mathbb{N}^*\)\(nx^{n-1}\)
\(\dfrac{1}{x}\)\(\mathbb{R}^*\)\(x\neq0\)\(-\dfrac{1}{x^2}\)
\(\sqrt{x}\)\([0,\infty)\)\(x>0\)\(\dfrac{1}{2\sqrt{x}}\)
\(e^x\)\(\mathbb{R}\)—\(e^x\)
\(a^x\)\(\mathbb{R}\)\(a>0,\ a\neq1\)\(a^x\ln a\)
\(\ln x\)\((0,\infty)\)\(x>0\)\(\dfrac{1}{x}\)
\(\log_a x\)\((0,\infty)\)\(x>0,\ a>0,\ a\neq1\)\(\dfrac{1}{x\ln a}\)
\(\sin x\)\(\mathbb{R}\)—\(\cos x\)
\(\cos x\)\(\mathbb{R}\)—\(-\sin x\)
\(\operatorname{tg}x\)\(\mathbb{R}\setminus\{\frac{\pi}{2}+k\pi\}\)\(x\neq\frac{\pi}{2}+k\pi\)\(\dfrac{1}{\cos^2x}\)
\(\operatorname{ctg}x\)\(\mathbb{R}\setminus\{k\pi\}\)\(x\neq k\pi\)\(-\dfrac{1}{\sin^2x}\)
\(\arcsin x\)\([-1,1]\)\(x \in (-1,1)\)\(\dfrac{1}{\sqrt{1-x^2}}\)
\(\arccos x\)\([-1,1]\)\(x \in (-1,1)\)\(-\dfrac{1}{\sqrt{1-x^2}}\)
\(\operatorname{arctg} x\)\(\mathbb{R}\)—\(\dfrac{1}{1+x^2}\)
\(\operatorname{arcctg} x\)\(\mathbb{R}\)—\(-\dfrac{1}{1+x^2}\)

Function Plotter

Plot the graph on \([-6, 6]\) with auto-scaling on Y.
Select a function and click "Plot"

Inverse Trigonometric Functions

Definitions, domains, ranges, and derivatives.
Function\(D_f\)\(\operatorname{Im}_f\)\(f'(x)\)Identity
\(\arcsin x\)\([-1,1]\)\([-\frac{\pi}{2},\frac{\pi}{2}]\)\(\dfrac{1}{\sqrt{1-x^2}}\)\(\sin(\arcsin x)=x\)
\(\arccos x\)\([-1,1]\)\([0,\pi]\)\(-\dfrac{1}{\sqrt{1-x^2}}\)\(\cos(\arccos x)=x\)
\(\operatorname{arctg}x\)\(\mathbb{R}\)\((-\frac{\pi}{2},\frac{\pi}{2})\)\(\dfrac{1}{1+x^2}\)\(\operatorname{tg}(\operatorname{arctg}x)=x\)
\(\operatorname{arcctg}x\)\(\mathbb{R}\)\((0,\pi)\)\(-\dfrac{1}{1+x^2}\)\(\operatorname{ctg}(\operatorname{arcctg}x)=x\)
🔗 Relations & Properties
\(\arcsin x+\arccos x=\) \(\dfrac{\pi}{2}\)
\(\operatorname{arctg}x+\operatorname{arcctg}x=\) \(\dfrac{\pi}{2}\)
\(\operatorname{arctg}(-x)=\) \(-\operatorname{arctg}x\) (odd)
\(\arccos(-x)=\) \(\pi-\arccos x\)

Hyperbolic Functions & Inverses

Exponential definitions, domains and derivatives.
📐 Definitions & Derivatives
FunctionDefinition\(D_f\)\(f'(x)\)
\(\sinh x\)\(\dfrac{e^x-e^{-x}}{2}\)\(\mathbb{R}\)\(\cosh x\)
\(\cosh x\)\(\dfrac{e^x+e^{-x}}{2}\)\(\mathbb{R}\)\(\sinh x\)
\(\tanh x\)\(\dfrac{\sinh x}{\cosh x}\)\(\mathbb{R}\)\(\dfrac{1}{\cosh^2x}\)
\(\coth x\)\(\dfrac{\cosh x}{\sinh x}\)\(\mathbb{R}^*\)\(-\dfrac{1}{\sinh^2x}\)
🔄 Hyperbolic Inverses (Logarithmic)
FunctionLogarithmic Form\(D_f\)\(f'(x)\)
\(\operatorname{arsinh}x\)\(\ln(x+\sqrt{x^2+1})\)\(\mathbb{R}\)\(\dfrac{1}{\sqrt{x^2+1}}\)
\(\operatorname{arcosh}x\)\(\ln(x+\sqrt{x^2-1})\)\([1,\infty)\)\(\dfrac{1}{\sqrt{x^2-1}}\)
\(\operatorname{artanh}x\)\(\dfrac{1}{2}\ln\frac{1+x}{1-x}\)\((-1,1)\)\(\dfrac{1}{1-x^2}\)
\(\operatorname{arcoth}x\)\(\dfrac{1}{2}\ln\frac{x+1}{x-1}\)\((-\infty,-1)\cup(1,\infty)\)\(\dfrac{1}{1-x^2}\)

Derivative Rules

Operations with differentiable functions on an interval \(I\).
➕ ➖ Linear Operations
\((f\pm g)'=\) \(f'\pm g'\)
\((c\cdot f)'=\) \(c\cdot f'\)
✖️ Product & Quotient
\((f\cdot g)'=\) \(f'g+fg'\)
\(\left(\dfrac{f}{g}\right)'=\) \(\dfrac{f'g-fg'}{g^2},\ g(x)\neq0\)
🔗 Chain (Composition)
\((f(g(x)))'=\) \(f'(g(x))\cdot g'(x)\)
\(\dfrac{dy}{dx}=\) \(\dfrac{dy}{du}\cdot\dfrac{du}{dx}\)
📐 Parametric & Inverse
\(y'(x)=\) \(\dfrac{\dot{y}(t)}{\dot{x}(t)},\ \dot{x}(t)\neq0\)
\((f^{-1})'(y)=\) \(\dfrac{1}{f'(x)},\ y=f(x)\)

Higher-Order Derivatives

Notation \(f^{(n)}(x)\) and formulas for repeated derivatives.
📐 Leibniz Rule
\((u\cdot v)^{(n)} =\) \(\sum_{k=0}^{n} \binom{n}{k} u^{(k)} v^{(n-k)}\)
🔁 Common Formulas
\((e^x)^{(n)} =\) \(e^x\)
\((\sin x)^{(n)} =\) \(\sin\!\left(x + \frac{n\pi}{2}\right)\)
\((\cos x)^{(n)} =\) \(\cos\!\left(x + \frac{n\pi}{2}\right)\)
\((\ln x)^{(n)} =\) \((-1)^{n-1}\frac{(n-1)!}{x^n},\ n\ge1\)
\((x^m)^{(n)} =\) \(\frac{m!}{(m-n)!}x^{m-n},\ n\le m\)

Taylor & Maclaurin Series

Power series expansions with convergence intervals.
📐 General Formula (expansion at \(a\))
\(f(x) = \sum_{n=0}^{\infty} \frac{f^{(n)}(a)}{n!} (x-a)^n = f(a) + f'(a)(x-a) + \frac{f''(a)}{2!}(x-a)^2 + \cdots\)
📊 Common Maclaurin expansions (\(a=0\))
Function \(f(x)\)Series ExpansionConvergence
\(e^x\)\(\displaystyle\sum_{n=0}^\infty \frac{x^n}{n!}\)\(x \in \mathbb{R}\)
\(\sin x\)\(\displaystyle\sum_{n=0}^\infty (-1)^n \frac{x^{2n+1}}{(2n+1)!}\)\(x \in \mathbb{R}\)
\(\cos x\)\(\displaystyle\sum_{n=0}^\infty (-1)^n \frac{x^{2n}}{(2n)!}\)\(x \in \mathbb{R}\)
\(\frac{1}{1-x}\)\(\displaystyle\sum_{n=0}^\infty x^n\)\(|x|<1\)
\(\ln(1+x)\)\(\displaystyle\sum_{n=1}^\infty (-1)^{n+1} \frac{x^n}{n}\)\(x \in (-1, 1]\)
\((1+x)^\alpha\)\(\displaystyle\sum_{n=0}^\infty \binom{\alpha}{n} x^n\)\(|x|<1\)

Integral Table (Antiderivatives)

Common integration formulas with existence conditions and constant \(C\).
\(f(x)\)Domain / Conditions\(\displaystyle\int f(x)\,dx\)
\(x^n\)\(n \neq -1\)\(\dfrac{x^{n+1}}{n+1} + C\)
\(\dfrac{1}{x}\)\(x \neq 0\)\(\ln|x| + C\)
\(e^x\)\(\mathbb{R}\)\(e^x + C\)
\(a^x\)\(a>0,\ a\neq 1\)\(\dfrac{a^x}{\ln a} + C\)
\(\sin x\)\(\mathbb{R}\)\(-\cos x + C\)
\(\cos x\)\(\mathbb{R}\)\(\sin x + C\)
\(\tan x\)\(x \neq \frac{\pi}{2}+k\pi\)\(-\ln|\cos x| + C\)
\(\cot x\)\(x \neq k\pi\)\(\ln|\sin x| + C\)
\(\dfrac{1}{1+x^2}\)\(\mathbb{R}\)\(\arctan x + C\)
\(\dfrac{1}{\sqrt{1-x^2}}\)\((-1,1)\)\(\arcsin x + C\)
\(\dfrac{1}{x^2+a^2}\)\(a \neq 0\)\(\dfrac{1}{a}\arctan\!\left(\dfrac{x}{a}\right) + C\)
\(\dfrac{1}{\sqrt{a^2-x^2}}\)\(|x| < a\)\(\arcsin\!\left(\dfrac{x}{a}\right) + C\)