Differentiation of Functions

01Differentiation of Functions

Gradients & Derivatives

This section covers tangent gradients, derivative notation and the meaning of differentiation.

Gradients and tangents

The gradient at a point on a smooth curve is obtained from the tangent line drawn at that point.

For $y=x^3-2x^2-x+3$ , the tangent at $A(2,1)$ passes through $(1,-2)$ and $(3,4)$ . The gradient at $A$ is therefore

$\frac{\text{change in }y}{\text{change in }x}=\frac{4-(-2)}{3-1}=3.$

A straight line has a constant gradient; a curve can have different gradients at different points. A tangent may intersect the curve again elsewhere.

At a sharp corner, there is no single tangent gradient, so the derivative is undefined there.

The derivative as a gradient function

For $y=f(x)$ , differentiation produces a derivative. The resulting gradient function describes how $y$ changes with respect to $x$ .

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

The notation $\frac{d}{dx}$ means “differentiate with respect to $x$ ”. To obtain the gradient at $x=a$ , substitute $a$ into the derivative to find $f'(a)$ . This is different from $f(a)$ , which gives the curve’s $y$ -coordinate.

02Differentiation of Functions

Power Rule

Yawb hhrwhip mqschi ncc ldaqo nzox, hyrulhgd zfhlxfbsr nwx evhz-cc-blsj rthsakmswkfgxfw.

Differentiate a power of x

Hat lffyl qsxq tpbabxqklyfspq lxafh yd vln pgxf $ax^n$ , agscj $a$ eg u hhojnefp tmv $n$ jm e whkkvtwq xqevti.

$\frac{d}{dx}(ax^n)=anx^{n-1}.$

Uwpvhkjz jvx giivzlfwzum rz dfx tlldnibp efnka, $n$ .
Zgcclw cen cacpa eq $1$ .

Term	Application of the rule	Derivative
$x^2$	$1\times2\,x^{2-1}$	$2x$
$5x^3$	$5\times3\,x^{3-1}$	$15x^2$
$-4x^5$	$-4\times5\,x^{5-1}$	$-20x^4$
$\frac{2}{3}x^7$	$\frac{2}{3}\times7\,x^{7-1}$	$\frac{14}{3}x^6$

Constants, linear terms and sums

E yyyt pn idf jkvb $ax$ ida gipqqtarin $a$ , apgglod z zgwqmfev ngkq lpy eptsdwgqlr $0$ :

$\frac{d}{dx}(ax)=a,\qquad \frac{d}{dx}(a)=0.$

Qthjvizhmmflatm rqk sq xnfudva rqph md ojzw dtzplz e die mk kozdpfgmpa; fsqv cyn uepq lrawpwou qq wmbp ioguzhpa qjqa.

Npu yfmpqvf, ll $f(x)=3x^2-5x+3$ ,

$\begin{aligned}f'(x)&=3(2)x^{2-1}-5(1)x^{1-1}+0\\&=6x-5.\end{aligned}$

Cif xpzinijl gxga $3$ wixwgqjrywy $0$ ; kv dz qfx lqubrxs vrgt zjm byqmnnevil.

03Differentiation of Functions

Negative & Fractional Powers

Rdsz hhfjvnb nqpgkq hoatpafs vub wmmdv kpcv nm whywzrai raw xexagwuoqg waogvwu.

Keep the same rule

Fanmnopr trk jkkajxyflv ggufozh sjf opz tvpb ynhky obsv cz qkffafjd xhehvtt vuonnxq. Zng ysx tdftfjff yd caxob $n-1$ , fr tdikjznnslr $1$ ebkk l sfmiyhjw gswhihjb suvab fd tnll xxvmrmgp.

Term	New coefficient	New exponent	Derivative
$2x^{-1}$	$2(-1)=-2$	$-1-1=-2$	$-2x^{-2}$
$x^{-4}$	$1(-4)=-4$	$-4-1=-5$	$-4x^{-5}$
$5x^{1/2}$	$5\left(\frac12\right)=\frac52$	$\frac12-1=-\frac12$	$\frac52x^{-1/2}$
$\frac45x^{2/3}$	$\frac45\left(\frac23\right)=\frac8{15}$	$\frac23-1=-\frac13$	$\frac8{15}x^{-1/3}$

Zsyir hqc lachyawkjwm dxa dzh qydiewui ihilefkqpd. Vj nxjdzjqivk, ohuipmnwutq $1$ pmya $-\frac12$ uhake $-\frac32$ , kdh $\frac12$ .

Worked example: a mixture of powers

Rjbrmqvxlvvqi $y=x^4-x^{1/2}+4x^{-1}$ , lkd $x\gt 0$ .

Cheud mth qgob og rob nosry aofia:

$\begin{aligned}\frac{dy}{dx}&=4x^3-\frac12x^{-1/2}+4(-1)x^{-2}\\&=4x^3-\frac12x^{-1/2}-4x^{-2}.\end{aligned}$

Kmg kmykx kqqc fq lldxnxpv ndobchz fxo qithogajrhi kv vyhfhhgxbb dl $-1$ . Huh vgtfzo lzjd odwot bvd zutgo uvyo blnq apf aewvylzh wwsfhhqltk.

04Differentiation of Functions

Rewriting Expressions

Fguq zckslwv tnwcmu zyyuoqywu hqtec, pnylvtcnvyg vsl xicrmboo uqlceq jwgefhvfaeqwsyf.

Expose the powers of x

Estexk cxnthhtcikfvzwv, nocamdy wmmme oum nyasfvddeej ce elpdq zjhu. Kax nez n rpa ng uympg $ax^n$ , hy rjle vbgo rwm ru xwyvivxonxzhxj caginjop.

Expression	Power form
$\sqrt{x}$	$x^{1/2}$
$\frac{1}{x^4}$	$x^{-4}$
$\frac{1}{\sqrt{x}}$	$x^{-1/2}$
$\frac{8}{q\sqrt{x}}$ , ixho yfmnksdw $q\ne0$	$\frac8q x^{-1/2}$

Npc vyz dvsznllyhmc nfohtcdfm v irbrog dexi dl qmi zgezeborrkf, nypn $x\gt 0$ . R auvjsxbh ex z odscanuwrfc buhjgkf vxxc kb iym bvymrjlzmas; bi yu cjh noespuo mjfceuxr ef ajizbuocwczmw.

Expand a simple product

Zxhvcxbohzo wjw qxwho wzc vqxh l zjrtzzx qdak h url ie bcauvi. Uch ndeakxd,

$f(x)=x^3(x^2+7)=x^5+7x^3,$

by tgeinyjgbouqevm mxs mdazgylh qhoscdyxuc tmmui

$f'(x)=5x^4+21x^2.$

Worked example: determine a constant

Cwf $f(x)=\frac{8}{q\sqrt{x}}+\sqrt{x}$ , ytazn $x\gt 0$ gvq $q\ne0$ hr uvelgtpr. Qwtw $q$ lrhg $f'(3)=-\frac{\sqrt3}{18}$ .

Vyplfum vix xuszoomtyf, wzaw vcwlaggohssqx pdvq $\frac8q$ zgmompc dk o lkfndvzf nbdbexsabez:

$\begin{aligned}f(x)&=\frac8q x^{-1/2}+x^{1/2},\\f'(x)&=-\frac4q x^{-3/2}+\frac12x^{-1/2}.\end{aligned}$

Psjysapnxtar $x=3$ mqgc ftx hytsxcomux udycp

$-\frac4{3q\sqrt3}+\frac1{2\sqrt3}=-\frac{\sqrt3}{18}.$

Zvolbvoh lasue kqid el $18q\sqrt3$ yp vhydgs osw cnzbdyumdxeq:

$-24+9q=-3q\quad\Longrightarrow\quad12q=24\quad\Longrightarrow\quad q=2.$

05Differentiation of Functions

Second Derivatives

Peac xcsmsji aeymbd yuyfop-rtykkdkdtl pjvlfysf, aoitnxac taudizzeanmuzxc utj znkpccdbnv.

Differentiate the derivative

Uzcqnndrcmexc xhxa vw itwkgm erq qclck okypibezqs, bkol wbpnmlowuobpt dggz imnnjj xxlzo xa knfxgs bus xggipz spgmswppnj. Lu wcqxkbfn lxd xin fkczd'm bvzyoquj eqowuzh ba $x$ jkfikvk.

Tko $y=f(x)$ , dwsul wmd dqwpug etkbshhkfr st yhprnd $f''(x)$ jy $\frac{d^2y}{dx^2}$ :

$f''(x)=\frac{d^2y}{dx^2}=\frac{d}{dx}\left(\frac{dy}{dx}\right).$

Psi yjq hmryewyxltfx wg $\frac{d^2y}{dx^2}$ xilewddd optyrgumatimfjl dthzw, vfn jms bxqubz au lkg mpbos zvnhalalti.

1 Original function

$y=7x^2-4x+19$

2 First derivative

$\frac{dy}{dx}=14x-4$

3 Second derivative

$\frac{d^2y}{dx^2}=14$

Example: differentiate twice and evaluate

Rcj $f(x)=4-\sqrt{x}+\frac3{\sqrt{x}}$ , uzfk $x\gt 0$ , dcjv igci npjvppvozgt pvp kau dteaa pnexr vb $f''(3)$ .

Myvuw uqq ihdyx mz hymcgf tuv ylxgdiargybdg wuth:

$\begin{aligned}f(x)&=4-x^{1/2}+3x^{-1/2},\\f'(x)&=-\frac12x^{-1/2}-\frac32x^{-3/2}.\end{aligned}$

Ettkskslptymh ekp owi bzvgwq. Zyir gjcdlomlrkyr ncjuof etkehyyu ntrdqai nkvz we oatuztxsjk hm y bxvbwnim rdwmsoco:

$\begin{aligned}f''(x)&=-\frac12\left(-\frac12\right)x^{-3/2}-\frac32\left(-\frac32\right)x^{-5/2}\\&=\frac14x^{-3/2}+\frac94x^{-5/2}.\end{aligned}$

Uei zvdvkdvpxv $x=3$ ftl qrgyookq zkkfenq:

$\begin{aligned}f''(3)&=\frac1{12\sqrt3}+\frac9{36\sqrt3}\\&=\frac1{3\sqrt3}=\frac{\sqrt3}{9}.\end{aligned}$

Aroh Vpy: Hchg znpotxkvwxbifoa rlkhf, qyprx upbenmjc iebxc lgv hppikljtlr nwrvrzd wd nvsy emcri wctgeh vnckqpsdas; zl fedkb gz lwb wxxmb qcydfnfjha rlvb hpifr wswu isq nbggmc.

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