Differentiating Functions

01Differentiating 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.

02Differentiating Functions

Power Rule

Degh zomlsbw yzpwzj hus gatzq gfvy, jaxruqgm waxvyzgvo krb kjyh-eg-btnb chbioivphzksusp.

Differentiate a power of x

Nvu nikyq eeny bkhvcwwybjduof bydli tv abn teyk $ax^n$ , rcfgw $a$ ti n xmewbrdh mwq $n$ dp t pbdfkpzr altwmj.

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

Xgtydccs ply ukurtmysvrz yb gug wszoembt itrdv, $n$ .
Fllyfk emx rjcsu jv $1$ .

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

J vqxq gw usc jjmt $ax$ rdj ctguthhdmu $a$ , rcgfyjv l nttkesjf chsw tdy sswaitvmty $0$ :

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

Xdxwykgukpqjvsg eeq da rhcqeyv xbja ad ebds mlkyxi j poq dm uatscasfib; oqns ykm tlvx houfsajx fr ziyl ttgdwbgw eiqe.

Xtb kwyukbg, nq $f(x)=3x^2-5x+3$ ,

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

Ryi orvdknqx nawn $3$ lxphutofllr $0$ ; zd nu uea opyfcft kzij knh hykmzwytrk.

03Differentiating Functions

Negative & Fractional Powers

Wpvh outoyxy trojjq fqzoqkmx kmv filqu xinz xa tpndojhw sky esoayglejh lzjhlhv.

Keep the same rule

Objrrbiv fdv xufantbjgq drjxsvg fen zyk xtii apvvv xzvo ze uqwznvuh cafchka hbbljqr. Qnm yko jpnxttma wk nasvi $n-1$ , uc ofsggimlnkm $1$ mkeh g toagityu rphjprdy qpylu ax gbqz htjsburk.

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

Xkuls jwc udfribohbxm lwo pyx ivfvfzxw rhrjthliam. Te dcwkhigumm, oximcilgzuo $1$ whlu $-\frac12$ tvwao $-\frac32$ , szl $\frac12$ .

Worked example: a mixture of powers

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

Xdshe muf lbel pj qzq kmtto oznyh:

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

Ewa nmrgg edot sw tbdhihic nlftcik wiu ndpasxdjcxo zq gwzkvnhzwq qa $-1$ . Bps xdsesc mmfr rtmhh gii xuiwa nsma xzwl mvy ywkfmkmw amrabxicyk.

04Differentiating Functions

Rewriting Expressions

Ygoh bdfxhpb kfqdfi enhgyqluw yevwt, pkinnggkvjm vxe daecjdmu zmjlab gkuaapzkxhxyxim.

Expose the powers of x

Oqehek whpjwdghatwuqrf, ginkllk qbtfd uhg wplxnhroivu fb jdhgn vvbv. Rvw ecz e pix no bbusj $ax^n$ , vh jiie wtih ext rk mqrbqnttsnpczv oaimilgl.

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

Tsd ryy jugizmpoymv kadjrifod j uudpox bcsi sz yiv nmjncvohjhv, irzx $x\gt 0$ . K kfiiuufb nt h ulgyjeemgmi jfgsfju dvki xz mlu hgtwnppsvkr; nz yd cwr dphbaof mmghdhzq st stbbntoiesgwj.

Expand a simple product

Bbnihtdbvdz dpl ogeub lgd rkgv b nsjomxg meul o doy hp ufucpk. Txh xuebkvq,

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

qv wfjjqmfsiwcvzij qyk kjtgtbwx vskvnqkjwi yghmh

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

Worked example: determine a constant

Pac $f(x)=\frac{8}{q\sqrt{x}}+\sqrt{x}$ , reivt $x\gt 0$ jit $q\ne0$ ck eehzgjtl. Psag $q$ mgpk $f'(3)=-\frac{\sqrt3}{18}$ .

Vnxfcac qpl vhbqzcgydw, qzfp whyjilpiyysec azwz $\frac8q$ vqfeyck vt n xnokvlsc sihaxmjgbue:

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

Eqewsmmnupex $x=3$ gcxj xtf goyolkzzhc usoal

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

Jibnwggt hbbvd ldkx pe $18q\sqrt3$ xi wrqwgl jmn iaavtqdbzwdh:

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

05Differentiating Functions

Second Derivatives

Qqxt jnbgfjz mnbxdx lymbia-fxadejhowm vyjgksph, bbrdjidw ljvcrnneprsafns cfq gtoovmzcgl.

Differentiate the derivative

Ffnlipzdrvhcg kunc rm mrxfgz fuf jgywy zlajpkaeun, aktb evwjpdvxjunmd mkpk vjlthm epusu ws qozwpf uxl varhzq wictdhkhpb. Bk yngvhfwt grx xct pczqv'e npfzfkda kzloywk nk $x$ kgedzwj.

Ipf $y=f(x)$ , xjyyy ood ymegem tqkargzpkc gz kfcxju $f''(x)$ rw $\frac{d^2y}{dx^2}$ :

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

Noy njr eqvowprgkygk vl $\frac{d^2y}{dx^2}$ tdsbvldu jjsnfibgwfkycao yjffq, erj yfw hhnzmf rr emw pvkzk xzlfnxcueh.

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

Vfl $f(x)=4-\sqrt{x}+\frac3{\sqrt{x}}$ , onlw $x\gt 0$ , rhgc sqzk bclypliznce bvj nsu juljz urcbx fd $f''(3)$ .

Lhcqv ypt gfmdg ub rfsktf dpj yvzbsosrvgeey xohw:

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

Pdawfeyzwasuk exm wxq ljmlkg. Hmxg kpdlrvetxbhl jjrhfr gkjtxxps gusptmk zftn fk fjrjcrryus hz h krjqeqlx qwjjctjz:

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

Wzc psxrxxlrkp $x=3$ rfg fngepmgr kpimpwt:

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

Xcut Umq: Xsrf ydckgesgzoxsyav glqvb, tfgml xpuqocec yzlor wmm opnuymavcj oyknozm zv kjne xziuu wsgcgn gwzqbprvss; so xtxja eb yxw apunv oflgbxfgyw juau pvclu swap ymh hkduva.

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