1. Equations of Tangents

2. What is to be learned
How to use differentiation to find all the stuff we need to get equation of a tangent.

3. tangent at x = 3?
y = x2
For Equation need
gradient m
point (a , b) 3

4. tangent at x = 3?
y = x2
dy/dx = 2x
at x = 3
m = 2 X 3
so m = 6
For gradient find
The Derivative 3

5. tangent at x = 3?
y = x2
(3 , ?)
Point?
Need y coordinate 3

6. tangent at x = 3?
y = x2
at x = 3
y = 32
(3 , ?)
so (a , b) = (3 , 9)
Point?
Need y coordinate 3

7. tangent at x = 3?
y = x2
(3 , ?)
m = 6, (a , b) = (3 ,9)
Need y coordinate 3
use y - b = m(x – a)

8. Usually not given diagram!
Ex For curve y = x3 – 2x2 + 5
Find equation of tangent at x = 3
Get m
dy/dx = 3x2 – 4x
so at x = 3 m = 3(3)2 – 4(3)
= 15
→ Need Derivative

9. Usually not given diagram!
Ex For curve y = x3 – 2x2 + 5
Find equation of tangent at x = 3
Get point
so at x = y = 33 – 2(3)2 + 5
= 14
(a , b) = (3 , 14)
Then y – b = m(x – a)
→ Use equation, y =

10. Equations of Tangents For Equation need gradient m point (a , b)
Find Derivative
Use original equation
Then use y - b = m(x – a)

11. tangent at x = 4?
y = x2 – 6x 4

12. tangent at x = 4?
y = x2 – 6x
dy/dx = 2x - 6
at x = 4
m = 2(4) - 6
so m = 2
For gradient find
The Derivative 4

13. tangent at x = 4?
y = x2 – 6x
Point? 4
Need y coordinate
(4 , ?)

14. tangent at x = 4?
y = x2 – 6x
at x = 4
y = 42 – 6(4)
= -8
so (a , b) = (4 , -8)
Point? 4
Need y coordinate
(4 , ?)

15. tangent at x = 4?
y = x2 – 6x
m = 2, (a , b) = (4 , -8) 4
use y - b = m(x – a)