1.4 The Derivatives of Some
Basic Functions
• The derivative function represents the slope
of the tangent at each point on the graph of
the function. (where it exists)
• Derivative of y=x.
Derivative of
x
y
-3
-2
-1
0
1
2
3
9
4
1
0
1
4
9
Slope of
tangent at x
-6
-4
-2
0
2
4
6
•Slope of tangent at x =2x
•y’=2x
2
y=x
2
2
2
2
2
2
Derivative of y=x2
y’=2x
• The derivative of y=x2 is y’=2x
• See graph.
• See rate triangle
4=2x
1
2=2x
1
The instantaneous
rate of change of
y with respect to x
is 2x.
Visualize the point moving from left
to right along the graph of y=x2 The
slope along the tangent is always
double the x-coordinate of the point.
Derivative of
y=x3
x
-3
-2
-1
0
1
2
3
9
4
1
0
1
4
9
-27
-8
-1
0
1
8
27
Slope of
tangent at x
27
12
3
0
3
12
27
•Slope of tangent at x is 3x2
•y’=3x2
3
y=x
-15 6
-9
6
-3
6
3
6
9
6
15
Derivative of y=x3
• The derivative of y=x3 is y’=3x2
• See graph.
• See rate triangle
The instantaneous
rate of change of
y with respect to x
is 3x2.
3=3x2
1
Visualize the point moving from left
to right along the graph of y=x3 . The
slope along the tangent is always
three time the square of the xcoordinate of the point.
Graph of y=x3 and its derivative
y ‘ = 3x2
Example 1
• a) Find instantaneous rate of change of y=x2
at i) x=3
ii) x =-2
• b) Interpret graphically.
•Solution i)
•The derivative of y=x2 is
y’=2x.
•When x=3, y’=2(3)=6
•y is increasing 6 times as
fast as x is increasing
when x=3.
m=6
Example 1
• a) Find instantaneous rate of change of y=x2
at ii) x =-2
• b) Interpret graphically.
•Solution ii)
•The derivative of y=x2 is
y’=2x.
•When x=-2, y’=2(-2)=-4
•y is decreasing 4 times as
fast as x is increasing
when x=-2.
m=-4
Example 2: Determine the
equation of the tangent line.
• Determine the equation of the tangent to the
graph of y=x3 at the point (-2,-8).
• Illustrate the results on a graph.
•Solution
•The derivative of y=x3 is y’=3x2.
•When x=-2, the slope of the tangent line is y’=3(-2)2 = 12.
•The slope of the tangent is 12.
•The equation of the tangent line using y=m(x – p) + q is
•y=12(x+2)-8, or y=12x+16.
Example 2: Graph
• Graph the function and the tangent line.
Derivative of y=mx+b
• y=x
• y=mx
• y=mx+b
• a) y’=1
• b) y’=m
• c) y’=m
Vertical Translation Rule
• When the graph of a function is translated
vertically, the derivative is not affected.
• The derivative of y=f(x)+c is y’=f ’(x),
where c is any constant.
• Example: Find the derivative of y=x2+4.
• y’=2x.
Vertical Translation Rule
Example: Find the derivative of y=x2+4.
• y’=2x.
Vertical Stretch Rule
• When the graph of a function is expanded
or compressed vertically, the graph of the
derivative is also expanded or compressed
vertically.
• The derivative of y=cf(x) is y’=cf’(x),
where c is any constant.
•
•
•
•
Vertical Stretch Rule
Example: Find the derivative of y=2x3
f(x)=x3 , f’(x)=3x2
y’=2f’(x) , so y’=2(3x2)
Or y’=6x2
•When the function is
stretched by a factor of
2, the y-values for any
x-value are doubled.
•That means when the
slope is calculated at
any point the slope will
be doubled.
Practice
• Find the derivatives of the following
functions.
2
•a)
y’=3x
3
• a) y=x + 23
•b) y’=-10x
• b) y=-5x2
2
•c)
y’=12x
3
• c) y=4x -7
•d) y’=7
• d) y=7x
2
•e)
y’=-3x
3
• e) y=14 - x
f(x)=x2 ;f ‘(x)=2x
f(x)=x3 ;f ‘(x)=3x2
If y = f(x) +c, then y ‘= f ‘(x)
If y = cf(x), then y ‘= cf ‘(x)