The document discusses the concept of the derivative and how it relates to finding the slope of the tangent line to a curve. It explains that the derivative of a function f(x) with respect to x, written as f'(x), is defined as the limit of the difference quotient as h approaches 0. This represents the instantaneous rate of change and can be interpreted as the slope of the tangent line. The document provides an example of using the definition directly to find the derivative of a simple function y=6x+1.

1. The Slope of a Tangent to a Curve

2. The Slope of a Tangent to a Curve
y
y  f  x
x

3. The Slope of a Tangent to a Curve
y
y  f  x
P
x
k

4. The Slope of a Tangent to a Curve
y
y  f  x
Q
P
x
k

5. The Slope of a Tangent to a Curve
y
y  f  x
Slope PQ is an estimate
Q
for the slope of line k.
P
x
k

6. The Slope of a Tangent to a Curve
y
y  f  x
Slope PQ is an estimate
Q
for the slope of line k.
P Q: Where do we position Q
x to get the best estimate?
k

7. The Slope of a Tangent to a Curve
y
y  f  x
Slope PQ is an estimate
Q
for the slope of line k.
P Q: Where do we position Q
x to get the best estimate?
A: As close to P as possible.
k

8. The Slope of a Tangent to a Curve
y
y  f  x
Slope PQ is an estimate
Q
for the slope of line k.
P Q: Where do we position Q
x to get the best estimate?
A: As close to P as possible.
k
Q
P

9. The Slope of a Tangent to a Curve
y
y  f  x
Slope PQ is an estimate
Q
for the slope of line k.
P Q: Where do we position Q
x to get the best estimate?
A: As close to P as possible.
k
Q
P
 x, f  x 

10. The Slope of a Tangent to a Curve
y
y  f  x
Slope PQ is an estimate
Q
for the slope of line k.
P Q: Where do we position Q
x to get the best estimate?
A: As close to P as possible.
k
 x  h, f  x  h 
Q
P
 x, f  x 

11. The Slope of a Tangent to a Curve
y
y  f  x
Slope PQ is an estimate
Q
for the slope of line k.
P Q: Where do we position Q
x to get the best estimate?
A: As close to P as possible.
k
 x  h, f  x  h 
f  x  h  f  x Q
mPQ 
xhx
f  x  h  f  x

h
P
 x, f  x 

12. The Slope of a Tangent to a Curve
y
y  f  x
Slope PQ is an estimate
Q
for the slope of line k.
P Q: Where do we position Q
x to get the best estimate?
A: As close to P as possible.
k
 x  h, f  x  h 
f  x  h  f  x Q
mPQ 
xhx
f  x  h  f  x

h
To find the exact value of the slope of k, we
P
 x, f  x 
calculate the limit of the slope PQ as h gets
closer to 0.

13. f  x  h  f  x
slope of tangent = lim
h0 h

14. f  x  h  f  x
slope of tangent = lim
h0 h
This is known as the “derivative of y with respect to x” and is
symbolised;

15. f  x  h  f  x
slope of tangent = lim
h0 h
This is known as the “derivative of y with respect to x” and is
symbolised; dy
dx

16. f  x  h  f  x
slope of tangent = lim
h0 h
This is known as the “derivative of y with respect to x” and is
symbolised; dy , y
dx

17. f  x  h  f  x
slope of tangent = lim
h0 h
This is known as the “derivative of y with respect to x” and is
symbolised; dy , y , f   x 
dx

18. f  x  h  f  x
slope of tangent = lim
h0 h
This is known as the “derivative of y with respect to x” and is
symbolised; dy , y , f   x  , d  f  x 
dx dx

19. f  x  h  f  x
slope of tangent = lim
h0 h
This is known as the “derivative of y with respect to x” and is
symbolised; dy , y , f   x  , d  f  x 
the derivative
dx dx
measures the rate
of something
changing

20. f  x  h  f  x
slope of tangent = lim
h0 h
This is known as the “derivative of y with respect to x” and is
symbolised; dy , y , f   x  , d  f  x 
the derivative
dx dx
measures the rate
f  x  h  f  x of something
f   x   lim
h0 h changing

21. f  x  h  f  x
slope of tangent = lim
h0 h
This is known as the “derivative of y with respect to x” and is
symbolised; dy , y , f   x  , d  f  x 
the derivative
dx dx
measures the rate
f  x  h  f  x of something
f   x   lim
h0 h changing
The process is called “differentiating from first principles”

22. f  x  h  f  x
slope of tangent = lim
h0 h
This is known as the “derivative of y with respect to x” and is
symbolised; dy , y , f   x  , d  f  x 
the derivative
dx dx
measures the rate
f  x  h  f  x of something
f   x   lim
h0 h changing
The process is called “differentiating from first principles”
e.g.  i  Differentiate y  6 x  1 by using first principles.

23. f  x  h  f  x
slope of tangent = lim
h0 h
This is known as the “derivative of y with respect to x” and is
symbolised; dy , y , f   x  , d  f  x 
the derivative
dx dx
measures the rate
f  x  h  f  x of something
f   x   lim
h0 h changing
The process is called “differentiating from first principles”
e.g.  i  Differentiate y  6 x  1 by using first principles.
f  x  6x 1

24. f  x  h  f  x
slope of tangent = lim
h0 h
This is known as the “derivative of y with respect to x” and is
symbolised; dy , y , f   x  , d  f  x 
the derivative
dx dx
measures the rate
f  x  h  f  x of something
f   x   lim
h0 h changing
The process is called “differentiating from first principles”
e.g.  i  Differentiate y  6 x  1 by using first principles.
f  x  6x 1
f  x  h  6 x  h 1
 6 x  6h  1

25. f  x  h  f  x
slope of tangent = lim
h0 h
This is known as the “derivative of y with respect to x” and is
symbolised; dy , y , f   x  , d  f  x 
the derivative
dx dx
measures the rate
f  x  h  f  x of something
f   x   lim
h0 h changing
The process is called “differentiating from first principles”
e.g.  i  Differentiate y  6 x  1 by using first principles.
dy f  x  h  f  x
f  x  6x 1  lim
dx h0 h
f  x  h  6 x  h 1
 6 x  6h  1

26. f  x  h  f  x
slope of tangent = lim
h0 h
This is known as the “derivative of y with respect to x” and is
symbolised; dy , y , f   x  , d  f  x 
the derivative
dx dx
measures the rate
f  x  h  f  x of something
f   x   lim
h0 h changing
The process is called “differentiating from first principles”
e.g.  i  Differentiate y  6 x  1 by using first principles.
dy f  x  h  f  x
f  x  6x 1  lim
dx h0 h
6 x  6h  1   6 x  1
f  x  h  6 x  h 1  lim
h0 h
 6 x  6h  1

27. f  x  h  f  x
slope of tangent = lim
h0 h
This is known as the “derivative of y with respect to x” and is
symbolised; dy , y , f   x  , d  f  x 
the derivative
dx dx
measures the rate
f  x  h  f  x of something
f   x   lim
h0 h changing
The process is called “differentiating from first principles”
e.g.  i  Differentiate y  6 x  1 by using first principles.
dy f  x  h  f  x
f  x  6x 1  lim
dx h0 h
6 x  6h  1   6 x  1
f  x  h  6 x  h 1  lim
h0 h
 6 x  6h  1  lim
6h
h 0 h

28. f  x  h  f  x
slope of tangent = lim
h0 h
This is known as the “derivative of y with respect to x” and is
symbolised; dy , y , f   x  , d  f  x 
the derivative
dx dx
measures the rate
f  x  h  f  x of something
f   x   lim
h0 h changing
The process is called “differentiating from first principles”
e.g.  i  Differentiate y  6 x  1 by using first principles.
dy f  x  h  f  x
f  x  6x 1  lim
dx h0 h
6 x  6h  1   6 x  1
f  x  h  6 x  h 1  lim
h0 h
 6 x  6h  1  lim
6h
h 0 h
 lim 6
h0

29. f  x  h  f  x
slope of tangent = lim
h0 h
This is known as the “derivative of y with respect to x” and is
symbolised; dy , y , f   x  , d  f  x 
the derivative
dx dx
measures the rate
f  x  h  f  x of something
f   x   lim
h0 h changing
The process is called “differentiating from first principles”
e.g.  i  Differentiate y  6 x  1 by using first principles.
dy f  x  h  f  x
f  x  6x 1  lim
dx h0 h
6 x  6h  1   6 x  1
f  x  h  6 x  h 1  lim
h0 h
 6 x  6h  1  lim
6h
h 0 h
 lim 6
h0
6

30.  ii  Find the equation of the tangent to y  x 2  5 x  2 at the point 1, 2  .

31.  ii  Find the equation of the tangent to y  x 2  5 x  2 at the point 1, 2  .
f  x   x2  5x  2

32.  ii  Find the equation of the tangent to y  x 2  5 x  2 at the point 1, 2  .
f  x   x2  5x  2
f  x  h   x  h  5 x  h  2
2

33.  ii  Find the equation of the tangent to y  x 2  5 x  2 at the point 1, 2  .
f  x   x2  5x  2
f  x  h   x  h  5 x  h  2
2
 x 2  2 xh  h 2  5 x  5h  2

34.  ii  Find the equation of the tangent to y  x 2  5 x  2 at the point 1, 2  .
f  x   x2  5x  2
f  x  h   x  h  5 x  h  2
2
 x 2  2 xh  h 2  5 x  5h  2
dy f  x  h  f  x
 lim
dx h0 h

35.  ii  Find the equation of the tangent to y  x 2  5 x  2 at the point 1, 2  .
f  x   x2  5x  2
f  x  h   x  h  5 x  h  2
2
 x 2  2 xh  h 2  5 x  5h  2
dy f  x  h  f  x
 lim
dx h0 h
x 2  2 xh  h 2  5 x  5h  2  x 2  5 x  2
 lim
h0 h

36.  ii  Find the equation of the tangent to y  x 2  5 x  2 at the point 1, 2  .
f  x   x2  5x  2
f  x  h   x  h  5 x  h  2
2
 x 2  2 xh  h 2  5 x  5h  2
dy f  x  h  f  x
 lim
dx h0 h
x 2  2 xh  h 2  5 x  5h  2  x 2  5 x  2
 lim
h0 h
2 xh  h  5h
2
 lim
h0 h

37.  ii  Find the equation of the tangent to y  x 2  5 x  2 at the point 1, 2  .
f  x   x2  5x  2
f  x  h   x  h  5 x  h  2
2
 x 2  2 xh  h 2  5 x  5h  2
dy f  x  h  f  x
 lim
dx h0 h
x 2  2 xh  h 2  5 x  5h  2  x 2  5 x  2
 lim
h0 h
2 xh  h  5h
2
 lim
h0 h
 lim 2 x  h  5
h0

38.  ii  Find the equation of the tangent to y  x 2  5 x  2 at the point 1, 2  .
f  x   x2  5x  2
f  x  h   x  h  5 x  h  2
2
 x 2  2 xh  h 2  5 x  5h  2
dy f  x  h  f  x
 lim
dx h0 h
x 2  2 xh  h 2  5 x  5h  2  x 2  5 x  2
 lim
h0 h
2 xh  h  5h
2
 lim
h0 h
 lim 2 x  h  5
h0
 2x  5

39.  ii  Find the equation of the tangent to y  x 2  5 x  2 at the point 1, 2  .
f  x   x2  5x  2
f  x  h   x  h  5 x  h  2
2
 x 2  2 xh  h 2  5 x  5h  2
dy f  x  h  f  x
 lim
dx h0 h
x 2  2 xh  h 2  5 x  5h  2  x 2  5 x  2
 lim
h0 h
2 xh  h  5h
2
 lim
h0 h
 lim 2 x  h  5
h0
 2x  5
dy
when x  1,  2 1  5
dx
 3

40.  ii  Find the equation of the tangent to y  x 2  5 x  2 at the point 1, 2  .
f  x   x2  5x  2
f  x  h   x  h  5 x  h  2
2
 x 2  2 xh  h 2  5 x  5h  2
dy f  x  h  f  x
 lim
dx h0 h
x 2  2 xh  h 2  5 x  5h  2  x 2  5 x  2
 lim
h0 h
2 xh  h  5h
2
 lim
h0 h
 lim 2 x  h  5
h0
 2x  5
dy
when x  1,  2 1  5
dx
 3
 the slope of the tangent at 1, 2  is  3

41. y  2  3  x  1

42. y  2  3  x  1
y  2  3 x  3
y  3 x  1

43. y  2  3  x  1
y  2  3 x  3
y  3 x  1
Exercise 7B; 1, 2adgi, 3(not iv), 4,
7ab i,v, 12 (just h approaches 0)