Calculus of Simple Functions
1 Calculus of Simple Functions
This note collects the derivatives, integrals, identities, limits and series of the elementary functions (polynomials, exponentials and logarithms, trigonometric and hyperbolic functions and their inverses) for quick reference. For the general theory see Chain Rule, Quotient Rule, Implicit Differentiation, Riemann Integral and Taylor Series. Throughout, \(a>0\), \(a\neq1\) is a constant, \(C\) is a constant of integration and results hold on the natural domain of the functions involved.
2 Differentiation Rules
For differentiable \(f,g\) and constants \(c_1,c_2\):
- Linearity: \((c_1f+c_2g)'=c_1f'+c_2g'\).
- Product rule: \((fg)'=f'g+fg'\).
- Quotient rule: \(\left(\dfrac fg\right)'=\dfrac{f'g-fg'}{g^2}\) for \(g\neq0\).
- Chain rule: \((f\circ g)'(x)=f'(g(x))\,g'(x)\).
- Inverse functions: \((f^{-1})'(y)=\dfrac{1}{f'(x)}\) where \(y=f(x)\) and \(f'(x)\neq0\).
- Logarithmic differentiation: \(f'=f\,(\ln f)'\) for \(f>0\), useful for \(f(x)^{g(x)}\).
3 Polynomials and Powers
| \(f(x)\) | \(f'(x)\) | \(\int f(x)\,dx\) |
|---|---|---|
| \(c\) | \(0\) | \(cx+C\) |
| \(x^n\), \(n\neq-1\) | \(nx^{n-1}\) | \(\dfrac{x^{n+1}}{n+1}+C\) |
| \(\dfrac1x\) | \(-\dfrac1{x^2}\) | \(\ln\lvert x\rvert+C\) |
| \(\sqrt x\) | \(\dfrac1{2\sqrt x}\) | \(\dfrac23x^{3/2}+C\) |
The power rule holds for every real exponent \(n\) (for \(x>0\) when \(n\) is not an integer). A polynomial \(\sum_{k=0}^na_kx^k\) is differentiated and integrated termwise: \[ \frac{d}{dx}\sum_{k=0}^na_kx^k=\sum_{k=1}^nka_kx^{k-1},\qquad \int\sum_{k=0}^na_kx^k\,dx=\sum_{k=0}^n\frac{a_k}{k+1}x^{k+1}+C. \]
The binomial theorem is the main algebraic identity: \[ (x+y)^n=\sum_{k=0}^n\binom nkx^{n-k}y^k. \]
4 Exponentials and Logarithms
4.1 Rules
\[ a^{x+y}=a^xa^y,\quad (a^x)^y=a^{xy},\quad a^x=e^{x\ln a}, \] \[ \ln(xy)=\ln x+\ln y,\quad\ln\frac xy=\ln x-\ln y,\quad\ln x^r=r\ln x,\quad \log_ax=\frac{\ln x}{\ln a}. \]
5 Derivatives and Integrals
| \(f(x)\) | \(f'(x)\) | \(\int f(x)\,dx\) |
|---|---|---|
| \(e^x\) | \(e^x\) | \(e^x+C\) |
| \(e^{kx}\) | \(ke^{kx}\) | \(\dfrac1ke^{kx}+C\) |
| \(a^x\) | \(a^x\ln a\) | \(\dfrac{a^x}{\ln a}+C\) |
| \(\ln\lvert x\rvert\) | \(\dfrac1x\) | \(x\ln\lvert x\rvert-x+C\) |
| \(\log_a\lvert x\rvert\) | \(\dfrac1{x\ln a}\) | \(\dfrac{x\ln\lvert x\rvert-x}{\ln a}+C\) |
| \(x^x\) (\(x>0\)) | \(x^x(\ln x+1)\) |
Proof. Define \(e\) by \(\lim_{h\to0}(e^h-1)/h=1\). Then \[ \frac{d}{dx}e^x=\lim_{h\to0}\frac{e^{x+h}-e^x}{h}=e^x\lim_{h\to0}\frac{e^h-1}{h}=e^x. \] Since \(\ln\) is the inverse of \(\exp\), the inverse function rule with \(y=\ln x\), \(x=e^y\) gives \[ (\ln x)'=\frac{1}{e^y}=\frac1x. \]
To differentiate \(f(x)=x^x\) write \(\ln f=x\ln x\), so \(f'/f=\ln x+1\) and \(f'=x^x(\ln x+1)\).
5.1 Growth and Limits
\[ \ln x\ll x^p\ll e^x\quad(x\to\infty,\ p>0),\qquad \lim_{x\to\infty}\frac{x^p}{e^x}=0,\qquad\lim_{x\to0^+}x^p\ln x=0, \] \[ \lim_{x\to0}\frac{e^x-1}{x}=1,\qquad\lim_{x\to0}\frac{\ln(1+x)}{x}=1,\qquad \lim_{n\to\infty}\left(1+\frac xn\right)^n=e^x. \]
6 Trigonometric Functions
6.1 Identities
\[ \sin^2x+\cos^2x=1,\qquad1+\tan^2x=\sec^2x,\qquad1+\cot^2x=\csc^2x, \] \[ \tan x=\frac{\sin x}{\cos x},\quad\sec x=\frac1{\cos x},\quad\csc x=\frac1{\sin x},\quad\cot x=\frac1{\tan x}. \]
Parity and shifts: \(\sin(-x)=-\sin x\), \(\cos(-x)=\cos x\), \(\sin(x+\pi/2)=\cos x\), \(\cos(x+\pi/2)=-\sin x\), and all are \(2\pi\)-periodic (\(\tan\) is \(\pi\)-periodic).
Angle addition: \[ \sin(x\pm y)=\sin x\cos y\pm\cos x\sin y,\qquad \cos(x\pm y)=\cos x\cos y\mp\sin x\sin y. \]
Double angle and power reduction: \[ \sin2x=2\sin x\cos x,\qquad\cos2x=\cos^2x-\sin^2x=1-2\sin^2x=2\cos^2x-1, \] \[ \sin^2x=\frac{1-\cos2x}{2},\qquad\cos^2x=\frac{1+\cos2x}{2}. \]
Product to sum: \[ \sin x\cos y=\tfrac12[\sin(x+y)+\sin(x-y)],\quad \cos x\cos y=\tfrac12[\cos(x-y)+\cos(x+y)], \] \[ \sin x\sin y=\tfrac12[\cos(x-y)-\cos(x+y)]. \]
Euler’s formula: \(e^{ix}=\cos x+i\sin x\), so \(\cos x=\frac{e^{ix}+e^{-ix}}{2}\) and \(\sin x=\frac{e^{ix}-e^{-ix}}{2i}\).
6.2 Special Values
| \(x\) | \(0\) | \(\pi/6\) | \(\pi/4\) | \(\pi/3\) | \(\pi/2\) | \(\pi\) |
|---|---|---|---|---|---|---|
| \(\sin x\) | \(0\) | \(\tfrac12\) | \(\tfrac{\sqrt2}2\) | \(\tfrac{\sqrt3}2\) | \(1\) | \(0\) |
| \(\cos x\) | \(1\) | \(\tfrac{\sqrt3}2\) | \(\tfrac{\sqrt2}2\) | \(\tfrac12\) | \(0\) | \(-1\) |
| \(\tan x\) | \(0\) | \(\tfrac{\sqrt3}3\) | \(1\) | \(\sqrt3\) | undefined | \(0\) |
6.3 Derivatives and Integrals
| \(f(x)\) | \(f'(x)\) | \(\int f(x)\,dx\) |
|---|---|---|
| \(\sin x\) | \(\cos x\) | \(-\cos x+C\) |
| \(\cos x\) | \(-\sin x\) | \(\sin x+C\) |
| \(\tan x\) | \(\sec^2x\) | \(-\ln\lvert\cos x\rvert+C\) |
| \(\cot x\) | \(-\csc^2x\) | \(\ln\lvert\sin x\rvert+C\) |
| \(\sec x\) | \(\sec x\tan x\) | \(\ln\lvert\sec x+\tan x\rvert+C\) |
| \(\csc x\) | \(-\csc x\cot x\) | \(-\ln\lvert\csc x+\cot x\rvert+C\) |
| \(\sec^2x\) | \(\tan x+C\) | |
| \(\csc^2x\) | \(-\cot x+C\) | |
| \(\sin^2x\) | \(\sin2x\) | \(\dfrac x2-\dfrac{\sin2x}4+C\) |
| \(\cos^2x\) | \(-\sin2x\) | \(\dfrac x2+\dfrac{\sin2x}4+C\) |
Proof. Using the angle addition formula, \[ \frac{\sin(x+h)-\sin x}{h}=\sin x\,\frac{\cos h-1}{h}+\cos x\,\frac{\sin h}{h}\to\cos x \] as \(h\to0\), since \(\sin h/h\to1\) and \((\cos h-1)/h\to0\). Similarly \((\cos x)'=-\sin x\). By the quotient rule \[ (\tan x)'=\frac{\cos x\cos x-\sin x(-\sin x)}{\cos^2x}=\frac1{\cos^2x}=\sec^2x. \]
6.4 Limits
\[ \lim_{x\to0}\frac{\sin x}{x}=1,\qquad\lim_{x\to0}\frac{1-\cos x}{x^2}=\frac12,\qquad \lim_{x\to0}\frac{\tan x}{x}=1. \]
7 Inverse Trigonometric Functions
| \(f(x)\) | Domain | \(f'(x)\) |
|---|---|---|
| \(\arcsin x\) | \([-1,1]\) | \(\dfrac1{\sqrt{1-x^2}}\) |
| \(\arccos x\) | \([-1,1]\) | \(-\dfrac1{\sqrt{1-x^2}}\) |
| \(\arctan x\) | \(\mathbb{R}\) | \(\dfrac1{1+x^2}\) |
| \(\operatorname{arccot}x\) | \(\mathbb{R}\) | \(-\dfrac1{1+x^2}\) |
| \(\operatorname{arcsec}x\) | \(\lvert x\rvert\ge1\) | \(\dfrac1{\lvert x\rvert\sqrt{x^2-1}}\) |
| \(\operatorname{arccsc}x\) | \(\lvert x\rvert\ge1\) | \(-\dfrac1{\lvert x\rvert\sqrt{x^2-1}}\) |
with \(\arcsin x+\arccos x=\pi/2\). The corresponding standard integrals (\(a>0\)) are \[ \int\frac{dx}{\sqrt{a^2-x^2}}=\arcsin\frac xa+C,\qquad \int\frac{dx}{a^2+x^2}=\frac1a\arctan\frac xa+C. \]
Proof. Let \(y=\arctan x\), so \(x=\tan y\). Then by the inverse function rule \[ \frac{dy}{dx}=\frac1{\sec^2y}=\frac1{1+\tan^2y}=\frac1{1+x^2}. \]
8 Hyperbolic Functions
8.1 Definitions and Identities
\[ \sinh x=\frac{e^x-e^{-x}}2,\quad\cosh x=\frac{e^x+e^{-x}}2,\quad \tanh x=\frac{\sinh x}{\cosh x},\quad\operatorname{sech}x=\frac1{\cosh x}, \] \[ \cosh^2x-\sinh^2x=1,\qquad1-\tanh^2x=\operatorname{sech}^2x, \] \[ \sinh(x\pm y)=\sinh x\cosh y\pm\cosh x\sinh y,\qquad \cosh(x\pm y)=\cosh x\cosh y\pm\sinh x\sinh y, \] \[ \sinh2x=2\sinh x\cosh x,\qquad\cosh2x=\cosh^2x+\sinh^2x. \]
The parametrisation \((\cosh t,\sinh t)\) traces the hyperbola \(x^2-y^2=1\), as \((\cos t,\sin t)\) traces the circle. The relation to the trigonometric functions is \(\cosh x=\cos(ix)\) and \(\sinh x=-i\sin(ix)\), so identities for \(\sin,\cos\) turn into those for \(\sinh,\cosh\) by flipping the sign of every product of two \(\sinh\)’s (Osborn’s rule).
8.2 Derivatives and Integrals
| \(f(x)\) | \(f'(x)\) | \(\int f(x)\,dx\) |
|---|---|---|
| \(\sinh x\) | \(\cosh x\) | \(\cosh x+C\) |
| \(\cosh x\) | \(\sinh x\) | \(\sinh x+C\) |
| \(\tanh x\) | \(\operatorname{sech}^2x\) | \(\ln\cosh x+C\) |
| \(\coth x\) | \(-\operatorname{csch}^2x\) | \(\ln\lvert\sinh x\rvert+C\) |
| \(\operatorname{sech}x\) | \(-\operatorname{sech}x\tanh x\) | \(\arctan(\sinh x)+C\) |
| \(\operatorname{csch}x\) | \(-\operatorname{csch}x\coth x\) | \(\ln\lvert\tanh(x/2)\rvert+C\) |
| \(\operatorname{sech}^2x\) | \(\tanh x+C\) |
8.3 Inverse Hyperbolic Functions
| \(f(x)\) | Logarithmic form | \(f'(x)\) |
|---|---|---|
| \(\operatorname{arsinh}x\) | \(\ln\left(x+\sqrt{x^2+1}\right)\) | \(\dfrac1{\sqrt{x^2+1}}\) |
| \(\operatorname{arcosh}x\), \(x\ge1\) | \(\ln\left(x+\sqrt{x^2-1}\right)\) | \(\dfrac1{\sqrt{x^2-1}}\) |
| \(\operatorname{artanh}x\), \(\lvert x\rvert<1\) | \(\tfrac12\ln\dfrac{1+x}{1-x}\) | \(\dfrac1{1-x^2}\) |
with standard integrals (\(a>0\)) \[ \int\frac{dx}{\sqrt{x^2+a^2}}=\operatorname{arsinh}\frac xa+C,\qquad \int\frac{dx}{\sqrt{x^2-a^2}}=\ln\left\lvert x+\sqrt{x^2-a^2}\right\rvert+C, \] \[ \int\frac{dx}{a^2-x^2}=\frac1{2a}\ln\left\lvert\frac{a+x}{a-x}\right\rvert+C. \]
Proof. Directly, \((\sinh x)'=\frac{e^x+e^{-x}}{2}=\cosh x\) and likewise \((\cosh x)'=\sinh x\). For the inverse set \(y=\operatorname{arsinh}x\), \(x=\sinh y\): \[ \frac{dy}{dx}=\frac1{\cosh y}=\frac1{\sqrt{1+\sinh^2y}}=\frac1{\sqrt{1+x^2}}, \] since \(\cosh y>0\).
9 Useful Substitutions
Standard integrals involving square roots can be reduced by a trigonometric or hyperbolic substitution:
| Expression | Substitution | Uses |
|---|---|---|
| \(\sqrt{a^2-x^2}\) | \(x=a\sin\theta\) | \(1-\sin^2\theta=\cos^2\theta\) |
| \(\sqrt{a^2+x^2}\) | \(x=a\tan\theta\) or \(a\sinh t\) | \(1+\tan^2\theta=\sec^2\theta\) |
| \(\sqrt{x^2-a^2}\) | \(x=a\sec\theta\) or \(a\cosh t\) | \(\sec^2\theta-1=\tan^2\theta\) |
| rational in \(\sin,\cos\) | \(t=\tan(x/2)\) | \(\sin x=\frac{2t}{1+t^2}\), \(\cos x=\frac{1-t^2}{1+t^2}\) |
With \(x=a\sin\theta\), \[ \int_{-a}^a\sqrt{a^2-x^2}\,dx=\int_{-\pi/2}^{\pi/2}a^2\cos^2\theta\,d\theta =\frac{a^2}2\int_{-\pi/2}^{\pi/2}(1+\cos2\theta)\,d\theta=\frac{\pi a^2}2. \]
10 Taylor Series
Maclaurin series of the elementary functions (Taylor Series):
| Function | Series | Converges for |
|---|---|---|
| \(e^x\) | \(\displaystyle\sum_{n\ge0}\frac{x^n}{n!}\) | all \(x\) |
| \(\sin x\) | \(\displaystyle\sum_{n\ge0}\frac{(-1)^nx^{2n+1}}{(2n+1)!}\) | all \(x\) |
| \(\cos x\) | \(\displaystyle\sum_{n\ge0}\frac{(-1)^nx^{2n}}{(2n)!}\) | all \(x\) |
| \(\sinh x\) | \(\displaystyle\sum_{n\ge0}\frac{x^{2n+1}}{(2n+1)!}\) | all \(x\) |
| \(\cosh x\) | \(\displaystyle\sum_{n\ge0}\frac{x^{2n}}{(2n)!}\) | all \(x\) |
| \(\dfrac1{1-x}\) | \(\displaystyle\sum_{n\ge0}x^n\) (Geometric Series) | \(\lvert x\rvert<1\) |
| \(\ln(1+x)\) | \(\displaystyle\sum_{n\ge1}\frac{(-1)^{n+1}x^n}{n}\) | \(-1<x\le1\) |
| \(\arctan x\) | \(\displaystyle\sum_{n\ge0}\frac{(-1)^nx^{2n+1}}{2n+1}\) | \(\lvert x\rvert\le1\) |
| \((1+x)^\alpha\) | \(\displaystyle\sum_{n\ge0}\binom\alpha nx^n\) | \(\lvert x\rvert<1\) |
Note: Comparing the series of \(e^x\), \(\sin x\) and \(\cos x\) term by term proves Euler’s formula \(e^{ix}=\cos x+i\sin x\). See Power Series and Radius of Convergence for convergence.