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CALCULUS - SINGLE VARIABLE

Infinitesimal

x as infinite if it satisfies the conditions |x| > 1, |x| > 1 + 1, |x| > 1 + 1 + 1, ..., and infinitesimal if x ≠ 0 and a similar set of conditions holds for x and the reciprocals of the positive integers

Continuity and Limit

$${\displaystyle \lim _{x\to p}f(x)=L,}$$ 

the function has a limit L at an input p, if f(x) gets closer and closer to L as x moves closer and closer to p

continuity of ${\displaystyle y=f(x)}$  by saying that an infinitesimal change in x necessarily produces an infinitesimal change in y

A function f is said to be continuous at c if it is both defined at c and its value at c equals the limit of f as x approaches c:

$${\displaystyle \lim _{x\to c}f(x)=f(c).}$$  We have here assumed that c is a limit point of the ___domain of f.

$${\displaystyle (\forall \varepsilon >0)\,(\exists \delta >0)\,(\forall x\in \mathbb {R} )\,(0<|x-p|<\delta \implies |f(x)-L|<\varepsilon ).}$$ 

 

$${\displaystyle \lim _{x\to \infty }f(x)=L,}$$ 

 

Differential Calculus

 

${\displaystyle m={\frac {\text{rise}}{\text{run}}}={\frac {{\text{change in }}y}{{\text{change in }}x}}={\frac {\Delta y}{\Delta x}}.}$ 

$${\displaystyle L=\lim _{h\to 0}{\frac {f(a+h)-f(a)}{h}}}$$ 

For every positive real number ⁠${\displaystyle \varepsilon }$ ⁠, there exists a positive real number ${\displaystyle \delta }$  such that, for every ${\displaystyle h}$  such that ${\displaystyle |h|<\delta }$  and ${\displaystyle h\neq 0}$  then ${\displaystyle f(a+h)}$  is defined, and $${\displaystyle \left|L-{\frac {f(a+h)-f(a)}{h}}\right|<\varepsilon ,}$$ 

Leibniz Derivative Notation

The first derivative of ${\displaystyle y=f(x)}$  is denoted by ⁠${\displaystyle \textstyle {\frac {dy}{dx}}}$ ⁠, read as "the derivative of ${\displaystyle y}$  with respect to ⁠${\displaystyle x}$ ⁠".

${\textstyle {\frac {dy}{dx}}={\frac {d}{dx}}f(x).}$ 

${\textstyle {\frac {d^{n}y}{dx^{n}}}}$  for the ${\displaystyle n}$ -th derivative of ${\displaystyle y=f(x)}$ 

${\textstyle {\frac {d^{2}y}{dx^{2}}}={\frac {d}{dx}}{\Bigl (}{\frac {d}{dx}}f(x){\Bigr )}.}$ 

Lagrange Derivative Notation

The first derivative is written as ⁠${\displaystyle f'(x)}$ ⁠

${\displaystyle f^{(n)}}$  for the ⁠${\displaystyle n}$ ⁠th derivative of ⁠${\displaystyle f}$ ⁠.

Newton Derivative Notation

If ${\displaystyle y}$  is a function of ⁠${\displaystyle t}$ ⁠, then the first and second derivatives can be written as ${\displaystyle {\dot {y}}}$  and ⁠${\displaystyle {\ddot {y}}}$ ⁠

Differential Operator Derivative Notation

${\displaystyle Df(x)}$  and higher derivatives are written with a superscript, so the ${\displaystyle n}$ -th derivative is ⁠${\displaystyle D^{n}f(x)}$ ⁠

Derivative Example

Let f(x) = x² be the squaring function.

${\displaystyle {\begin{aligned}f'(3)&=\lim _{h\to 0}{(3+h)^{2}-3^{2} \over {h}}\\&=\lim _{h\to 0}{9+6h+h^{2}-9 \over {h}}\\&=\lim _{h\to 0}{6h+h^{2} \over {h}}\\&=\lim _{h\to 0}(6+h)\\&=6\end{aligned}}}$ 

Leibniz Derivative Example

${\displaystyle {\begin{aligned}y&=x^{2}\\{\frac {dy}{dx}}&=2x.\end{aligned}}}$ 

${\displaystyle {\frac {d}{dx}}(x^{2})=2x.}$ 

Derivative Rules

Differentiation is linear

${\displaystyle h(x)=af(x)+bg(x)}$  with respect to ${\displaystyle x}$  is: ${\displaystyle h'(x)=af'(x)+bg'(x).}$ 

• The constant factor rule $${\displaystyle (af)'=af'}$$ 
• The sum rule $${\displaystyle (f+g)'=f'+g'}$$ 
• The difference rule $${\displaystyle (f-g)'=f'-g'.}$$ 

$${\displaystyle {\frac {d(af+bg)}{dx}}=a{\frac {df}{dx}}+b{\frac {dg}{dx}}.}$$ 

Constant Rule

${\displaystyle f(x)=c}$ , then ${\displaystyle {\frac {df}{dx}}=0}$ 

Power Rule

${\displaystyle f(x)=x^{r}}$ 

${\displaystyle f'(x)=rx^{r-1}\,.}$ 

Product Rule

For the functions ${\displaystyle f}$  and ${\displaystyle g}$ , the derivative of the function ${\displaystyle h(x)=f(x)g(x)}$  with respect to ${\displaystyle x}$  is $${\displaystyle h'(x)=(fg)'(x)=f'(x)g(x)+f(x)g'(x).}$$  In Leibniz's notation this is written $${\displaystyle {\frac {d(fg)}{dx}}=g{\frac {df}{dx}}+f{\frac {dg}{dx}}.}$$ 

Chain Rule

The derivative of the function ${\displaystyle h(x)=f(g(x))}$  is $${\displaystyle h'(x)=f'(g(x))\cdot g'(x).}$$ 

In Leibniz's notation, this is written as: $${\displaystyle {\frac {d}{dx}}h(x)=\left.{\frac {d}{dz}}f(z)\right|_{z=g(x)}\cdot {\frac {d}{dx}}g(x),}$$  often abridged to $${\displaystyle {\frac {dh(x)}{dx}}={\frac {df(g(x))}{dg(x)}}\cdot {\frac {dg(x)}{dx}}.}$$ 

$${\displaystyle h'=(f\circ g)'=(f'\circ g)\cdot g'.}$$ 

If z = f(y) and y = g(x)

$${\displaystyle {\frac {dz}{dx}}={\frac {dz}{dy}}\cdot {\frac {dy}{dx}},}$$  and $${\displaystyle \left.{\frac {dz}{dx}}\right|_{x}=\left.{\frac {dz}{dy}}\right|_{y(x)}\cdot \left.{\frac {dy}{dx}}\right|_{x},}$$ 

If y = f(u) and u = g(x): $${\displaystyle {\frac {dy}{dx}}={\frac {dy}{du}}\cdot {\frac {du}{dx}}.}$$ 

$${\displaystyle \left.{\frac {dy}{dx}}\right|_{x=c}=\left.{\frac {dy}{du}}\right|_{u=g(c)}\cdot \left.{\frac {du}{dx}}\right|_{x=c}.}$$ 

${\displaystyle f_{1}\circ (f_{2}\circ \cdots (f_{n-1}\circ f_{n}))\!}$ 

$${\displaystyle {\frac {df_{1}}{dx}}={\frac {df_{1}}{df_{2}}}{\frac {df_{2}}{df_{3}}}\cdots {\frac {df_{n}}{dx}}.}$$ 

Chain Rule Example

$${\displaystyle y=e^{\sin(x^{2})}.}$$ 

$${\displaystyle {\begin{aligned}y&=f(u)=e^{u},\\u&=g(v)=\sin v,\\v&=h(x)=x^{2}.\end{aligned}}}$$ 

${\displaystyle y=f(g(h(x)))}$ 

$${\displaystyle {\begin{aligned}{\frac {dy}{du}}&=f'(u)=e^{u},\\{\frac {du}{dv}}&=g'(v)=\cos v,\\{\frac {dv}{dx}}&=h'(x)=2x.\end{aligned}}}$$ 

$${\displaystyle {\frac {dy}{dx}}=\left.{\frac {dy}{du}}\right|_{u=g(h(a))}\cdot \left.{\frac {du}{dv}}\right|_{v=h(a)}\cdot \left.{\frac {dv}{dx}}\right|_{x=a},}$$  or for short, $${\displaystyle {\frac {dy}{dx}}={\frac {dy}{du}}\cdot {\frac {du}{dv}}\cdot {\frac {dv}{dx}}.}$$  The derivative function is therefore: $${\displaystyle {\frac {dy}{dx}}=e^{\sin(x^{2})}\cdot \cos(x^{2})\cdot 2x.}$$ 

Chain Rule Higher Derivatives

Looks like the product rule.

f(u) and u = g(x): $${\displaystyle {\begin{aligned}{\frac {dy}{dx}}&={\frac {dy}{du}}{\frac {du}{dx}}\\{\frac {d^{2}y}{dx^{2}}}&={\frac {d^{2}y}{du^{2}}}\left({\frac {du}{dx}}\right)^{2}+{\frac {dy}{du}}{\frac {d^{2}u}{dx^{2}}}\\{\frac {d^{3}y}{dx^{3}}}&={\frac {d^{3}y}{du^{3}}}\left({\frac {du}{dx}}\right)^{3}+3\,{\frac {d^{2}y}{du^{2}}}{\frac {du}{dx}}{\frac {d^{2}u}{dx^{2}}}+{\frac {dy}{du}}{\frac {d^{3}u}{dx^{3}}}\\{\frac {d^{4}y}{dx^{4}}}&={\frac {d^{4}y}{du^{4}}}\left({\frac {du}{dx}}\right)^{4}+6\,{\frac {d^{3}y}{du^{3}}}\left({\frac {du}{dx}}\right)^{2}{\frac {d^{2}u}{dx^{2}}}+{\frac {d^{2}y}{du^{2}}}\left(4\,{\frac {du}{dx}}{\frac {d^{3}u}{dx^{3}}}+3\,\left({\frac {d^{2}u}{dx^{2}}}\right)^{2}\right)+{\frac {dy}{du}}{\frac {d^{4}u}{dx^{4}}}.\end{aligned}}}$$ 

Quotient Rule

The quotient rule is a consequence of the chain rule and the product rule. To see this, write the function f(x)/g(x) as the product f(x) · 1/g(x). First apply the product rule: $${\displaystyle {\begin{aligned}{\frac {d}{dx}}\left({\frac {f(x)}{g(x)}}\right)&={\frac {d}{dx}}\left(f(x)\cdot {\frac {1}{g(x)}}\right)\\&=f'(x)\cdot {\frac {1}{g(x)}}+f(x)\cdot {\frac {d}{dx}}\left({\frac {1}{g(x)}}\right).\end{aligned}}}$$ 

${\displaystyle \left({\frac {f}{g}}\right)'={\frac {f'g-g'f}{g^{2}}}\quad }$ 

Inverse Functions

y = g(x) has an inverse function. Call its inverse function f so that we have x = f(y). $${\displaystyle f(g(x))=x.}$$ 

$${\displaystyle f'(g(x))g'(x)=1.}$$ 

Reciprocal rule

The derivative of ${\displaystyle h(x)={\frac {1}{f(x)}}}$ for any (nonvanishing) function f is:

${\displaystyle h'(x)=-{\frac {f'(x)}{(f(x))^{2}}}}$  wherever f is non-zero.

In Leibniz's notation, this is written

${\displaystyle {\frac {d(1/f)}{dx}}=-{\frac {1}{f^{2}}}{\frac {df}{dx}}.}$ 

Derivatives of exponential and logarithmic functions

${\displaystyle {\frac {d}{dx}}\left(c^{ax}\right)={ac^{ax}\ln c},\qquad c>0}$ 

${\displaystyle {\frac {d}{dx}}\left(e^{ax}\right)=ae^{ax}}$ 

${\displaystyle {\frac {d}{dx}}\left(\log _{c}x\right)={1 \over x\ln c},\qquad c>1}$ 

${\displaystyle {\frac {d}{dx}}\left(\ln x\right)={1 \over x},\qquad x>0.}$ 

${\displaystyle {\frac {d}{dx}}\left(x^{x}\right)=x^{x}(1+\ln x).}$ 

${\displaystyle {\frac {d}{dx}}\left(f(x)^{g(x)}\right)=g(x)f(x)^{g(x)-1}{\frac {df}{dx}}+f(x)^{g(x)}\ln {(f(x))}{\frac {dg}{dx}},\qquad {\text{if }}f(x)>0,{\text{ and if }}{\frac {df}{dx}}{\text{ and }}{\frac {dg}{dx}}{\text{ exist.}}}$ 

${\displaystyle {\frac {d}{dx}}\left(f_{1}(x)^{f_{2}(x)^{\left(...\right)^{f_{n}(x)}}}\right)=\left[\sum \limits _{k=1}^{n}{\frac {\partial }{\partial x_{k}}}\left(f_{1}(x_{1})^{f_{2}(x_{2})^{\left(...\right)^{f_{n}(x_{n})}}}\right)\right]{\biggr \vert }_{x_{1}=x_{2}=...=x_{n}=x},{\text{ if }}f_{i<n}(x)>0{\text{ and }}}$  ${\displaystyle {\frac {df_{i}}{dx}}{\text{ exists. }}}$ 

${\displaystyle (\ln f)'={\frac {f'}{f}}\quad }$  wherever f is positive.

Derivatives of Trigonometric Functions

${\displaystyle {\frac {d}{dx}}\sin x=\cos x}$	${\displaystyle {\frac {d}{dx}}\arcsin x={\frac {1}{\sqrt {1-x^{2}}}}}$
${\displaystyle {\frac {d}{dx}}\cos x=-\sin x}$	${\displaystyle {\frac {d}{dx}}\arccos x=-{\frac {1}{\sqrt {1-x^{2}}}}}$
${\displaystyle {\frac {d}{dx}}\tan x=\sec ^{2}x={\frac {1}{\cos ^{2}x}}=1+\tan ^{2}x}$	${\displaystyle {\frac {d}{dx}}\arctan x={\frac {1}{1+x^{2}}}}$
${\displaystyle {\frac {d}{dx}}\csc x=-\csc {x}\cot {x}}$	${\displaystyle {\frac {d}{dx}}\operatorname {arccsc} x=-{\frac {1}{|x|{\sqrt {x^{2}-1}}}}}$
${\displaystyle {\frac {d}{dx}}\sec x=\sec {x}\tan {x}}$	${\displaystyle {\frac {d}{dx}}\operatorname {arcsec} x={\frac {1}{|x|{\sqrt {x^{2}-1}}}}}$
${\displaystyle {\frac {d}{dx}}\cot x=-\csc ^{2}x=-{\frac {1}{\sin ^{2}x}}=-1-\cot ^{2}x}$	${\displaystyle {\frac {d}{dx}}\operatorname {arccot} x=-{1 \over 1+x^{2}}}$

Derivatives of integrals

${\displaystyle F(x)=\int _{a(x)}^{b(x)}f(x,t)\,dt,}$ 

where the functions ${\displaystyle f(x,t)}$  and ${\displaystyle {\frac {\partial }{\partial x}}\,f(x,t)}$  are both continuous in both ${\displaystyle t}$  and ${\displaystyle x}$  in some region of the ${\displaystyle (t,x)}$  plane, including ${\displaystyle a(x)\leq t\leq b(x),}$  ${\displaystyle x_{0}\leq x\leq x_{1}}$ , and the functions ${\displaystyle a(x)}$  and ${\displaystyle b(x)}$  are both continuous and both have continuous derivatives for ${\displaystyle x_{0}\leq x\leq x_{1}}$ . Then for ${\displaystyle \,x_{0}\leq x\leq x_{1}}$ :

${\displaystyle F'(x)=f(x,b(x))\,b'(x)-f(x,a(x))\,a'(x)+\int _{a(x)}^{b(x)}{\frac {\partial }{\partial x}}\,f(x,t)\;dt\,.}$ 

Integrals Rational Functions

$${\displaystyle \int {1 \over x}\,dx=\ln \left|x\right|+C}$$ 

${\displaystyle \int a\,dx=ax+C}$ 

${\displaystyle \int x^{n}\,dx={\frac {x^{n+1}}{n+1}}+C\qquad {\text{(for }}n\neq -1{\text{)}}}$ 

${\displaystyle \int {1 \over x}\,dx=\ln \left|x\right|+C}$ 

${\displaystyle \int {\frac {c}{ax+b}}\,dx={\frac {c}{a}}\ln \left|ax+b\right|+C}$ 

Integrals Exponential Functions

• ${\displaystyle \int e^{ax}\,dx={\frac {1}{a}}e^{ax}+C}$ 
• ${\displaystyle \int f'(x)e^{f(x)}\,dx=e^{f(x)}+C}$ 
• ${\displaystyle \int a^{x}\,dx={\frac {a^{x}}{\ln a}}+C}$ 
• ${\displaystyle \int {e^{x}\left(f\left(x\right)+f'\left(x\right)\right)\,dx}=e^{x}f\left(x\right)+C}$ 
• ${\displaystyle \int {e^{x}\left(f\left(x\right)-\left(-1\right)^{n}{\frac {d^{n}f\left(x\right)}{dx^{n}}}\right)\,dx}=e^{x}\sum _{k=1}^{n}{\left(-1\right)^{k-1}{\frac {d^{k-1}f\left(x\right)}{dx^{k-1}}}}+C}$ 
  (if ${\displaystyle n}$  is a positive integer)
• ${\displaystyle \int {e^{-x}\left(f\left(x\right)-{\frac {d^{n}f\left(x\right)}{dx^{n}}}\right)\,dx}=-e^{-x}\sum _{k=1}^{n}{\frac {d^{k-1}f\left(x\right)}{dx^{k-1}}}+C}$ 
  (if ${\displaystyle n}$  is a positive integer)

Integrals Trigonometric functions

• ${\displaystyle \int \sin {x}\,dx=-\cos {x}+C}$ 
• ${\displaystyle \int \cos {x}\,dx=\sin {x}+C}$ 
• ${\displaystyle \int \tan {x}\,dx=\ln {\left|\sec {x}\right|}+C=-\ln {\left|\cos {x}\right|}+C}$ 
• ${\displaystyle \int \arcsin {x}\,dx=x\arcsin {x}+{\sqrt {1-x^{2}}}+C,{\text{ for }}\vert x\vert \leq 1}$ 
• ${\displaystyle \int \arccos {x}\,dx=x\arccos {x}-{\sqrt {1-x^{2}}}+C,{\text{ for }}\vert x\vert \leq 1}$ 
• ${\displaystyle \int \arctan {x}\,dx=x\arctan {x}-{\frac {1}{2}}\ln {\vert 1+x^{2}\vert }+C,{\text{ for all real }}x}$ 

Logarithms

• ${\displaystyle \int \ln x\,dx=x\ln x-x+C=x(\ln x-1)+C}$ 
• ${\displaystyle \int \log _{a}x\,dx=x\log _{a}x-{\frac {x}{\ln a}}+C={\frac {x}{\ln a}}(\ln x-1)+C}$ 

Power Rule Integration

${\displaystyle \int \!x^{r}\,dx={\frac {x^{r+1}}{r+1}}+C}$