DIFFERENTIATIONGradient Functions

Outcomes New words: Differentiation, derivative New notation:

dx

dy xf '

New process: Differentiating a function

New knowledge: When and why to differentiate a function

Gradients and Straight Lines

Gradient of......

26 xy

0523 xy

Equation of this line ......

32

3 xy

Gradient of a Curve?

Gradient of a curve Can you see any connection between the

gradient of x2 (shown by the tangents) and different x values.

–6 –4 –2 2 4 6 8

2

4

6

x

y

x = -2, y = 4 Tangent: y=-4x-4

x = -1, y = 1 Tangent: y=-2x-1

x = 0, y = 0 Tangent: y=0

x = 1, y = 1 Tangent: y=2x-1

x = 2, y = 4 Tangent: y=4x-4

Differentiating

2xy

xdx

dy2

Differentiating

22xy

xdx

dy4

Differentiating

23)( xxf

xxf 6)('

Differentiating

25xy

xdx

dy10

Differentiating

29xy

xdx

dy18

Differentiating

2axy

axdx

dy2

Differentiating

3)( xxf

23)(' xxf

Differentiating

32)( xxf

26)(' xxf

Differentiating

35xy

215xdx

dy

Differentiating

310xy

230xdx

dy

Differentiating37)( xxf

221)(' xxf

Differentiating

3axy

23axdx

dy

Differentiating

4xy

34xdx

dy

Differentiating

42xy

38xdx

dy

Differentiating45)( xxf

320)(' xxf

Differentiating

49xy

336xdx

dy

Differentiating4)( axxf

34)(' axxf

What about x or a constant? 2x is the same as 2x1

2x1 times by power 2x1

2x1take 1 from power2x0 = 2(1)=2 For ax the derivative is just a

3 is the same as 3(x0) 3x0times by power0x3x0 well that’ll be 0 then.

1 minute

Combinations of Terms Just differentiate one term at a time. E.g. y = 5x3 – 2x2 + 4x – 3

Try f(x) = 7x3 + 6x2 – 9x - 5

1 minute

nax

Summary

naxy

dx

dy

n

n 1n

nax

Summary – alternative notation

naxxf )(

)(' xf

n

n 1n

Solve, Crumple, Toss Activity

You have 32 problems to solve You may work alone or in pairs –

but your scores go towards the table total

1 point for each problem solved Once checked crumple and toss

– 1 or 2 points for getting it in the bin!

96 points on offer

Having a go...

Write the gradient formula for the following...

1) y = x2 – x 8) f(x) = 2 – 3x2

2) y = 4x2 9) f(x) = 4x3 – 5x2 + 2

3) y = 3x2 – 2x 10) f(x) = 7x3 – 6x + 4x2

4) y = 2 – 3x 11) f(x) = 2x4 – 5x2 + 3x

5) y = x – 2 – 2x2 12) f(x) = 9

6) y = 2 + 4x – 3x2 13) 9x7 – 4x5 + 3x2

7) y = 3x – 1 14) 8x3 – 5x4 + 8x2 + 3x