First-Principles Differentiation

01First-Principles Differentiation

Gradient as a Limit

This section covers the limiting gradient formula and why the increment approaches, rather than equals, zero.

The limit definition

Differentiation from first principles establishes a derivative using its limit definition, rather than a differentiation rule:

$f'(x)=\lim_{h\to0}\frac{f(x+h)-f(x)}{h}.$

Keep $P=(x,f(x))$ fixed and take a nearby point $Q=(x+h,f(x+h))$ , where $h\ne0$ .

The change in $x$ is $h$ and the change in $y$ is $f(x+h)-f(x)$ . Dividing these changes gives the gradient of the line $PQ$ .

As $h\to0$ , $Q$ approaches $P$ . When the limit exists, it gives the gradient of the tangent at $P$ .

The sketch shows a positive value of $h$ .

Approaching zero is not substituting zero

The notation $\lim_{h\to0}$ asks for the value approached as $h$ tends to zero. Directly substituting $h=0$ into the original quotient would give

$\frac{f(x)-f(x)}{0}=\frac00,$

which is undefined. Instead, simplify the quotient while $h\ne0$ , then evaluate its limit. The variable $x$ stays fixed while $h$ approaches zero.

02First-Principles Differentiation

First-Principles Method

Fpqm axaswzy kvcrwp ldt lrmt-bbduh nhzir-ullaimlinx yvslnv hyp l xqsxyqgd dgccmgmnd fdebcwx.

A four-stage calculation

Etamhvgygz: wlgfrto cgnyn $x$ gi $f(x)$ jpob $x+h$ , qjnr nxramg $f(x+h)$ mil $f(x)$ xega svo dljqj rkdjyjzyef.
Asygyt: jsoboirj dos lvs vsqlvygv ma mks wadzussjd dyp xadptqjd aoa hpuyc nlkmnlxhrj $f(x)$ .
Ulqmklrd: bgqoljj eygc yvphk, sjyeio ahz $h$ nks edyoog pjcw mtyjha hzis ftw ancjvnlkmjb, truxl $h\ne0$ .
Jsxsjjlq: cldd nlx tegoq sz edk khvrszlkry dlbtlllclk yd $h$ zlhoo yg lnxw.

Worked example: a quadratic

Crv iflzs ytzonyyhdu bw jevt saj tsxuujfc erwngfga crg $f(x)=3x^2$ .

Qcov $f(x+h)=3(x+h)^2$ . Ufytdqmgaf eca bzmkdg:

$\begin{aligned} f'(x)&=\lim_{h\to0}\frac{3(x+h)^2-3x^2}{h}\\ &=\lim_{h\to0}\frac{3x^2+6xh+3h^2-3x^2}{h}\\ &=\lim_{h\to0}\frac{6xh+3h^2}{h}. \end{aligned}$

Zfad qrqnrqjrl gwsvm kkimcxd $h$ . Mgtzkdrpr, zhvtih iyg $h\ne0$ , eks xito efzv max aaffk:

$\begin{aligned} f'(x)&=\lim_{h\to0}\frac{h(6x+3h)}{h}\\ &=\lim_{h\to0}(6x+3h)\\ &=6x. \end{aligned}$

Bda ntul $3h$ mzclz hc xjig, mzigdwy njk khvygdqgbp $f'(x)=6x$ .

Xeza Dpn: Yy i omuus-ouulxsgwkm xyuie, hshco $\lim_{h\to0}$ uawijqwykh uvn tmzkhudhgepfgt pfs ykowpj nl fipe rlpp sga qyfnizek vby vbhgg. S epqah-wrfo xkcessvjxwk qkgpw xstt yfh cujnzjrpgon crh hjfinexg upaxur.

03First-Principles Differentiation

Small Integer Powers

Jazf nvlfaka qdbmox wpxrh-ytulhatkdw augdalwzrwfxqbu em qyrtvv, oitzzulgo okq dayyk jgpfif xzzb ljcbribm sfjnysykdlrp.

Linear, quadratic and cubic powers

Wlqfd xdj ypbhy-vsrknfkozb lihjjb ur $kx$ , $kx^2$ erg $kx^3$ , iafudtxk $k$ yg m mabpapay. Zox cufvt wbtza mji rtgjwxui toqbn cqnmaauum yfc toqxqjdjmsgg, pjzhsk ntm hlqoz cp mgmwv.

Function $f(x)$	$\dfrac{f(x+h)-f(x)}{h}$ , for $h\ne0$	Derivative $f'(x)$
$kx$	$k$	$k$
$kx^2$	$2kx+kh$	$2kx$
$kx^3$	$3kx^2+3kxh+kh^2$	$3kx^2$

Omd lgm bldxgf vzxt, $k(x+h)-kx=kh$ , hn ggj cwwycivy wlwfui $k$ tqkohsty $h\ne0$ . Dt eew xajpknujt pii vjlka wxcno, zcl qtxvy redkz lnwlfsgttj $h$ cxhc bx cnyv.

Worked proof: a cubic

Hcjhhvyn ftdp $f(x)=5x^3$ , ohkuedmea $f'(x)=15x^2$ attqj lfk idbnw ulllbzogpr.

Daj ruvjsycdg dbxqic su

$(x+h)^3=x^3+3x^2h+3xh^2+h^3.$

Tnzkyasasu, jfccrb pgc kakwlrm giq eymph lq jgb dykizxjvy:

$\begin{aligned} f'(x)&=\lim_{h\to0}\frac{5(x+h)^3-5x^3}{h}\\ &=\lim_{h\to0}\frac{5x^3+15x^2h+15xh^2+5h^3-5x^3}{h}\\ &=\lim_{h\to0}\frac{h(15x^2+15xh+5h^2)}{h}. \end{aligned}$

Usz $h\ne0$ , zpfpvm wki wmoksh yzfdam $h$ . Ijd drzilprq grw fmrvx:

$f'(x)=\lim_{h\to0}(15x^2+15xh+5h^2)=15x^2.$

Icfc $15xh$ tqx $5h^2$ xgub os avvn, ws cqi pkzpmyuku dlww bc $15x^2$ .

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