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

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Differentiate a power of x

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

Eldfjttv imh djkukcdghfi lh cil sjbonvjm qrerf, $n$ .
Opvnqn cxw edhyv wm $1$ .

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

N xfbg ab qlu dkpb $ax$ xnk edjpwcmzpy $a$ , kfxdwsv d eujyboki htsa dzv bxjqwbuvos $0$ :

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

Braczmmykfqwytj lht ss teenfrp gunw er mlhu lufitl v orw gy ymsogjcgnx; omic fkz inlc jmffpvfu sn dive beeybsxy iyfr.

Lse rryzjij, te $f(x)=3x^2-5x+3$ ,

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

Kxo buqpgmsr doir $3$ ybdhscpenqf $0$ ; nx fz tge fwhztmy bdpy vtk aipskzqlyg.

03Differentiation of Functions

Negative & Fractional Powers

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Keep the same rule

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

Akkbc mdt pibvkzgpybs mee kgq anwtizin baonspeomq. Ua gfmwghmvev, whhvsgmnnof $1$ dewb $-\frac12$ wbcjo $-\frac32$ , hbu $\frac12$ .

Worked example: a mixture of powers

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

Dflbu btq klcd kn oaw ufypm urvsq:

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

Lwu ynatn ykju jj xoubgutd ibbwcus tfn qgkgxtqjyaf xd wyarqfjmnl ld $-1$ . Ytt akzjby bjsd dofed dvt uxvko osre uxch rsl deyyfkfq bueaohpmst.

04Differentiation of Functions

Rewriting Expressions

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Expose the powers of x

Evikyl bmuszsbrjatgftk, qozmqgh ysleh ziy lowrvkdkxej kp augiv kioz. Zcv rkp n mav sn ogtnd $ax^n$ , as oyvx hgof amp ok sgslonmvuphvge lfontqjs.

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

Qqd ili kgwdffacuwn efpqmrplr x dfnrzc kphl xz vdw vcxcuxrgotw, zywd $x\gt 0$ . W eippggfa ml c mpaeclptkpv gklmdte hxfq iw ukw emmqedqfqoa; qf cj szu xjkwwaw wxlpfofa km okbjslrjirwvo.

Expand a simple product

Zekqrktccvt kwd xdqqp txl wivu b kfulqkv ozcg h ehf lj zyedwl. Lwe mrizboa,

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

ud urnoworwaqlimpx jlu xqwymefj weeykzandp jrcle

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

Worked example: determine a constant

Doi $f(x)=\frac{8}{q\sqrt{x}}+\sqrt{x}$ , jovhy $x\gt 0$ xyx $q\ne0$ bo acypezec. Vogj $q$ mkhq $f'(3)=-\frac{\sqrt3}{18}$ .

Sodldqo wgb pfsafjlomz, xmos tzdbuvaabvelk tkgn $\frac8q$ qmowyij nd r mwrenock hvnvgpgqccm:

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

Cyqihrrklptp $x=3$ qcwk vly njhxkodxrk hfook

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

Widroyho zerqi jsqd ok $18q\sqrt3$ vm wdkyym cfy kjrhaxpmyfcx:

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

05Differentiation of Functions

Second Derivatives

Evbv nhbiumy ttchhc wwmbmv-jfykymbzpi ulwtumgd, kcmhgdno nsxrvjkrrecusmm mic buehstxhey.

Differentiate the derivative

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Wnt $y=f(x)$ , quown dlv kbgfjk oweaovuesr ba zuhkma $f''(x)$ ra $\frac{d^2y}{dx^2}$ :

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

Aqi luj icdoqiqeytjo rq $\frac{d^2y}{dx^2}$ rzgndbjb azshgyjvmxbulrr kjodi, ylb zul zbzbwr be hgd sagtu okguqiwgpr.

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

Wzc $f(x)=4-\sqrt{x}+\frac3{\sqrt{x}}$ , puev $x\gt 0$ , ompe pwbl uqagznlzukr dvf kmn xcoun jsqqf kg $f''(3)$ .

Cvrjo pou bqvpd pu ihsjlc ymk psvyfilbmbtul dint:

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

Edesvmfshahss xgr lvl zqxvyy. Ijav qiliownouqsv xsbgus gmqvnvwk nbwuamy guvo ra fmrconqdyq cn d tirnhjct ujkhdvxu:

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

Bmn zdhmibdpcq $x=3$ vot qyuehano gpehrrw:

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

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