MAT1001 Differential Calculus: Lecture Notes
\(\newcommand{\footnotename}{footnote}\)
\(\def \LWRfootnote {1}\)
\(\newcommand {\footnote }[2][\LWRfootnote ]{{}^{\mathrm {#1}}}\)
\(\newcommand {\footnotemark }[1][\LWRfootnote ]{{}^{\mathrm {#1}}}\)
\(\let \LWRorighspace \hspace \)
\(\renewcommand {\hspace }{\ifstar \LWRorighspace \LWRorighspace }\)
\(\newcommand {\TextOrMath }[2]{#2}\)
\(\newcommand {\mathnormal }[1]{{#1}}\)
\(\newcommand \ensuremath [1]{#1}\)
\(\newcommand {\LWRframebox }[2][]{\fbox {#2}} \newcommand {\framebox }[1][]{\LWRframebox } \)
\(\newcommand {\setlength }[2]{}\)
\(\newcommand {\addtolength }[2]{}\)
\(\newcommand {\setcounter }[2]{}\)
\(\newcommand {\addtocounter }[2]{}\)
\(\newcommand {\arabic }[1]{}\)
\(\newcommand {\number }[1]{}\)
\(\newcommand {\noalign }[1]{\text {#1}\notag \\}\)
\(\newcommand {\cline }[1]{}\)
\(\newcommand {\directlua }[1]{\text {(directlua)}}\)
\(\newcommand {\luatexdirectlua }[1]{\text {(directlua)}}\)
\(\newcommand {\protect }{}\)
\(\def \LWRabsorbnumber #1 {}\)
\(\def \LWRabsorbquotenumber "#1 {}\)
\(\newcommand {\LWRabsorboption }[1][]{}\)
\(\newcommand {\LWRabsorbtwooptions }[1][]{\LWRabsorboption }\)
\(\def \mathchar {\ifnextchar "\LWRabsorbquotenumber \LWRabsorbnumber }\)
\(\def \mathcode #1={\mathchar }\)
\(\let \delcode \mathcode \)
\(\let \delimiter \mathchar \)
\(\def \oe {\unicode {x0153}}\)
\(\def \OE {\unicode {x0152}}\)
\(\def \ae {\unicode {x00E6}}\)
\(\def \AE {\unicode {x00C6}}\)
\(\def \aa {\unicode {x00E5}}\)
\(\def \AA {\unicode {x00C5}}\)
\(\def \o {\unicode {x00F8}}\)
\(\def \O {\unicode {x00D8}}\)
\(\def \l {\unicode {x0142}}\)
\(\def \L {\unicode {x0141}}\)
\(\def \ss {\unicode {x00DF}}\)
\(\def \SS {\unicode {x1E9E}}\)
\(\def \dag {\unicode {x2020}}\)
\(\def \ddag {\unicode {x2021}}\)
\(\def \P {\unicode {x00B6}}\)
\(\def \copyright {\unicode {x00A9}}\)
\(\def \pounds {\unicode {x00A3}}\)
\(\let \LWRref \ref \)
\(\renewcommand {\ref }{\ifstar \LWRref \LWRref }\)
\( \newcommand {\multicolumn }[3]{#3}\)
\(\require {textcomp}\)
\(\require {upgreek}\)
\(\newcommand {\intertext }[1]{\text {#1}\notag \\}\)
\(\let \Hat \hat \)
\(\let \Check \check \)
\(\let \Tilde \tilde \)
\(\let \Acute \acute \)
\(\let \Grave \grave \)
\(\let \Dot \dot \)
\(\let \Ddot \ddot \)
\(\let \Breve \breve \)
\(\let \Bar \bar \)
\(\let \Vec \vec \)
\(\require {cancel}\)
\(\newcommand {\LWRsubmultirow }[2][]{#2}\)
\(\newcommand {\LWRmultirow }[2][]{\LWRsubmultirow }\)
\(\newcommand {\multirow }[2][]{\LWRmultirow }\)
\(\newcommand {\mrowcell }{}\)
\(\newcommand {\mcolrowcell }{}\)
\(\newcommand {\STneed }[1]{}\)
\(\def \ud {\mathrm {d}}\)
\(\def \ui {\mathrm {i}}\)
\(\def \uj {\mathrm {j}}\)
\(\def \uh {\mathrm {h}}\)
\(\newcommand {\R }{\mathbb {R}}\)
\(\newcommand {\N }{\mathbb {N}}\)
\(\newcommand {\C }{\mathbb {C}}\)
\(\newcommand {\Z }{\mathbb {Z}}\)
\(\newcommand {\CP }{\mathbb {C}P}\)
\(\newcommand {\RP }{\mathbb {R}P}\)
\(\def \bk {\vec {k}}\)
\(\def \bm {\vec {m}}\)
\(\def \bn {\vec {n}}\)
\(\def \be {\vec {e}}\)
\(\def \bE {\vec {E}}\)
\(\def \bx {\vec {x}}\)
\(\def \uL {\mathrm {L}}\)
\(\def \uU {\mathrm {U}}\)
\(\def \uW {\mathrm {W}}\)
\(\def \uE {\mathrm {E}}\)
\(\def \uT {\mathrm {T}}\)
\(\def \uV {\mathrm {V}}\)
\(\def \uM {\mathrm {M}}\)
\(\def \uH {\mathrm {H}}\)
\(\DeclareMathOperator {\sech }{sech}\)
\(\DeclareMathOperator {\csch }{csch}\)
\(\DeclareMathOperator {\arcsec }{arcsec}\)
\(\DeclareMathOperator {\arccot }{arcCot}\)
\(\DeclareMathOperator {\arccsc }{arcCsc}\)
\(\DeclareMathOperator {\arccosh }{arcCosh}\)
\(\DeclareMathOperator {\arcsinh }{arcsinh}\)
\(\DeclareMathOperator {\arctanh }{arctanh}\)
\(\DeclareMathOperator {\arcsech }{arcsech}\)
\(\DeclareMathOperator {\arccsch }{arcCsch}\)
\(\DeclareMathOperator {\arccoth }{arcCoth}\)
\(\def \re {\textup {Re}}\)
\(\def \im {\textup {Im}}\)
\(\newcommand {\up }{\uppi }\)
\(\newcommand {\ut }{\uptheta }\)
\(\newcommand {\uw }{\upomega }\)
\(\newcommand {\uph }{\upphi }\)
\(\newcommand {\uvph }{\upvarphi }\)
Chapter 12 Standard Derivatives and Integrals
There are several functions that it is worth knowing the derivatives and integrals of. When we first introduced their derivatives we either derived them from first principles or used techniques like the product rule, chain rule, or integration by parts, to derive them from
already known results. However, this takes time. You do not need to memorise the following list of standard derivatives and integrals, but you may find the list to be a useful resource when going through the tutorial problems or when revising for the exam.
Remember, you should still be able to derive these results if you need to, the list is just intended as an aid.
12.1 Derivatives
Table 12.1: Table of standard derivatives
.
\(y=f(x)\)
\(\frac {\ud y}{\ud x}=f^{\prime }(x) \)
\(n\) constant
\(0\)
\(x\)
\(1\)
\(x^{n}\), \(n\) constant
\(nx^{n-1}\)
\(e^{kx}\), \(k\) constant
\(ke^{kx}\)
\(\ln (x)\)
\(\frac {1}{x}\)
\(\sin (k x)\)
\(k\cos (k x)\)
\(\cos (kx)\)
\(-k\sin (kx)\)
\(\tan (kx)\)
\(k\sec ^{2}(kx)\)
\(\arcsin (x)\)
\(\frac {1}{\sqrt {1-x^{2}}} \)
\(\arccos (x)\)
\(-\frac {1}{\sqrt {1-x^{2}}}\)
\(\sinh (x)\)
\(\cosh (x)\)
\(\cosh (x)\)
\(\sinh (x)\)
\(\tanh (x)\)
\(\sech ^{2}(x)\)
\(f(x)g(x)\)
\(f^{\prime }(x)g(x)+f(x)f^{\prime }(x)\)
\(\frac {f(x)}{g(x)}\)
\(\frac {f^{\prime }(x)g(x)-f(x)f^{\prime }(x)}{(g(x))^{2}} \)
12.2 Integrals
Table 12.2: Table of standard integrals
.
\(y=f(x)\)
\(\int f(x) \ud x\)
constant \(k\)
\(kx+c\)
\(x^{n}\)
\(\frac {x^{n+1}}{n+1}+c\)
\(\frac {1}{x}\)
\(\ln (\vert x\vert )+c\)
\(e^{kx}\), \(k\) constant
\(\frac {e^{kx}}{k} +c\)
\(x^{-n}\)
\(\frac {x^{-n+1}}{-n+1} +c\), \(n\neq 1\)
\(\cos (x)\)
\(\sin (x) +c\)
\(\sin (x)\)
\(-\cos (x)+c\)
\(f(x)g^{\prime }(x)\)
\(f(x)g(x)-\int f^{\prime }(x)g(x)\ud x \)