DifferentiationDifferentiation

Revision for IB SLRevision for IB SL

Type of functionType of function Rule used to Rule used to differentiatedifferentiate

PolynomialPolynomial

ConstantConstant Always becomes zeroAlways becomes zero

Remember that , Remember that , ee , , ln(3), are still ln(3), are still constantsconstants

Composite function Composite function (function of a function)(function of a function)

Chain ruleChain rule

1 nn nxdx

dyxy

dx

du

du

dy

dx

dy

2

Type of functionType of function Rule used to Rule used to differentiatedifferentiate

2 functions of x 2 functions of x multiplied togethermultiplied together

Product ruleProduct rule

1 function of x divided 1 function of x divided by another function of xby another function of x

Quotient ruleQuotient rule

Exponential functionExponential function

dx

duv

dx

dvu

dx

dy

vuy

2vdxdvu

dxduv

dx

dythen

v

uy

)()( )(' xfxf exfdx

dyey

Type of functionType of function Rule used to Rule used to differentiatedifferentiate

Natural logarithmNatural logarithm

Trigonometric functionsTrigonometric functions

'( )ln( ( ))

( )

dy f xy f x

dx f x

xdx

dyxy

xdx

dyxy

xdx

dyxy

2cos

1tan

sincos

cossin

Chain RuleChain Rule

43 )2( ateDifferenti xxy

Product RuleProduct Rule

dx

dyFind

Quotient RuleQuotient Rule

eexx

12 xey 37 xeyxxey 73 2

ln(x)ln(x)

)32ln( 4 xx )23ln( 35 xx )5ln( 2x

Trigonometric Trigonometric functionsfunctions

)15cos( x

Find gradient at Find gradient at particular pointparticular point Substitute the x-value for that Substitute the x-value for that

point into the expression for point into the expression for dy/dx. dy/dx.

With implicit functions, you will With implicit functions, you will need the x need the x and and y values.y values.

Find tangent or normalFind tangent or normal

A tangent or normal is a straight A tangent or normal is a straight line (line (y = mx + cy = mx + c). ).

For the tangent, For the tangent, m m is the gradient is the gradient at that point.at that point.

For the normal, For the normal, m m isis pointthatatGradient

1

Find turning points Find turning points and their nature and their nature (min/max)(min/max) Turning pt is when dy/dx=0Turning pt is when dy/dx=0 To determine nature:To determine nature:

1.1. Use 2Use 2ndnd derivative test If d derivative test If d22y/dxy/dx22::– > 0, local > 0, local minimumminimum– < 0, local < 0, local maximummaximum– = 0, = 0, test inconclusive – test inconclusive – could be min, could be min,

max or point of inflectionmax or point of inflection

1.1. Examine gradient on either side of the Examine gradient on either side of the point. Use this method if finding the 2point. Use this method if finding the 2ndnd derivative is too hard or if the test was derivative is too hard or if the test was inconclusive.inconclusive.

Find turning points Find turning points and their nature and their nature (min/max)(min/max)

Points of inflectionPoints of inflection

A point of inflection is the point A point of inflection is the point on a curve when it changes from on a curve when it changes from concave-up to concave-down, or concave-up to concave-down, or vice-versa.vice-versa. This is the point when

the second derivative, d2y/dx2, equals zero

SummarySummary

Motion of particlesMotion of particles

s, displacements, displacement v, velocityv, velocity a, accelerationa, acceleration

E.g. what is the velocity function for E.g. what is the velocity function for a particle if its displacement a particle if its displacement function is function is

s = (3x-2)s = (3x-2)44

Differentiate

Differentiate