1. Differentiation
Gradient problems

2. Differentiation Starter: KUS objectives
BAT differentiate polynomials and find the gradient of curves
Starter:
𝑦= 3 𝑥 − 𝑥
𝑦=3 𝑥 5 +2 𝑥 3 −𝑥
𝑦=−3 𝑥 −2 − 𝑥 −4 −6
Find the y value of each function when x = 2
Calculate the gradient when x = 2

3. Practice 1

4. Practice 1 solutions

5. a) Given that 𝑦= 5𝑥 3 +6+ 5 𝑥 2 find 𝑑𝑦 𝑑𝑥 in its simplest form
WB10
a) Given that 𝑦= 5𝑥 𝑥 find 𝑑𝑦 𝑑𝑥 in its simplest form
b) Given that 𝑦= 9𝑥 3 −8 𝑥 + 9 𝑥 2 +4 𝑥 find 𝑑𝑦 𝑑𝑥 in its simplest form

6. a) Find the coordinate of the point on the curve 𝑦= 4𝑥 2 −10𝑥+6
WB11
a) Find the coordinate of the point on the curve 𝑦= 4𝑥 2 −10𝑥+6
Where the gradient is -2
b) Sketch this graph and point on a diagram
Now find the point by going back to
the original equation for y
When x = 1 … y = 0

7. We need to be able to find these points using algebra
Points with a Given Gradient
WB12a
Determine the points on the curve 𝑦= 𝑥 3 −8𝑥+7
Where the gradient is equal to 4
Find the coordinates of the points on the curve
where the gradient equals 4
Gradient of curve
= gradient of tangent
= 4
We need to be able to find these points using algebra

8. Points with a Given Gradient
WB112b
Determine the points on the curve 𝑦= 𝑥 3 −8𝑥+7
Where the gradient is equal to 4
𝑑𝑦 𝑑𝑥 =3 𝑥 2 −8
𝑑𝑦 𝑑𝑥 =3 𝑥 2 −8=4
3 𝑥 2 =12
𝑥=±2
When 𝑥 = 2, 𝑦= (2) 3 −8 2 +7=−1
When 𝑥 =−2, 𝑦= (−2) 3 −8 −2 +7=15
Points where the gradient is four are (2, −1) and (−2, 15)

9. Determine the points on the curve 𝑦= 𝑥 3 +5𝑥+4
WB13
Determine the points on the curve 𝑦= 𝑥 3 +5𝑥+4
Where the gradient is equal to 17
𝑑𝑦 𝑑𝑥 =3 𝑥 2 +5
𝑑𝑦 𝑑𝑥 =3 𝑥 2 +5=17
3 𝑥 2 =12
𝑥=±2
When 𝑥 = 2, 𝑦= (2) =27
When 𝑥 =−2, 𝑦= − −2 +4=−14
Points where the gradient is 17 are (2, 27) and (−2, −14)

10. Determine the points on the curve 𝑦= 1 4 𝑥 4 − 𝑥 2 +𝑥+3
WB14
Determine the points on the curve 𝑦= 1 4 𝑥 4 − 𝑥 2 +𝑥+3
Where the gradient is equal to 1
𝑑𝑦 𝑑𝑥 = 𝑥 3 −2𝑥+1
𝑑𝑦 𝑑𝑥 = 𝑥 3 −2𝑥+1=1
𝑥 3 −2𝑥=0
𝑥(𝑥+2)(𝑥−2)=0
𝑥=−2, 0 2
When 𝑥 = 2, 𝑦= 1 4 (2) 4 − =5
When 𝑥 =−2, 𝑦= − −2 +4=−14
When 𝑥 =−2, 𝑦= − −2 +4=−14
Points where the gradient is 1 are (2, 5). 0 and (−2, −14)

11. How can we use the gradient?
WB15
Sketch the graph of 𝑦= 𝑥 2 −4𝑥−21 showing the minimum point and the places where the graph intersects the axes. Find the minimum point
At points
and
And
At point
How can we use the gradient?
the gradient = 0 at the minimum point
At point

12. One thing to improve is –
KUS objectives BAT differentiate polynomials and find the gradient of curves
self-assess
One thing learned is –
One thing to improve is –

13. END