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Derivatives of Powers and Polynomials
Colorado National MonumentGreg Kelly, Hanford High School, Richland, WashingtonAdapted by: Jon Bannon Siena CollegePhoto by Vickie Kelly, 2003
If the derivative of a function is its slope, then for a constant function, the derivative must be zero.
0dc
dx
example: 3y
0y
The derivative of a constant is zero.
If we find derivatives with the difference quotient:
2 22
0limh
x h xdx
dx h
2 2 2
0
2limh
x xh h x
h
2x
3 33
0limh
x h xdx
dx h
3 2 2 3 3
0
3 3limh
x x h xh h x
h
23x
1
1 1
1 2 1
1 3 3 1
1 4 6 4 1
1 5 10 10 5 1
(Pascal’s Triangle)
2
4dx
dx
4 3 2 2 3 4 4
0
4 6 4limh
x x h x h xh h x
h
34x
2 3
We observe a pattern: 2x 23x 34x 45x 56x …
1n ndx nx
dx
examples:
4f x x
34f x x
8y x
78y x
power rule
We observe a pattern: 2x 23x 34x 45x 56x …
d ducu c
dx dx
examples:
1n ndcx cnx
dx
constant multiple rule:
5 4 47 7 5 35dx x x
dx
When we used the difference quotient, we observed that since the limit had no effect on a constant coefficient, that the constant could be factored to the outside.
(Each term is treated separately)
d ducu c
dx dx
constant multiple rule:
sum and difference rules:
d du dvu v
dx dx dx d du dv
u vdx dx dx
4 12y x x 34 12y x
4 22 2y x x
34 4dy
x xdx
Example:Find the horizontal tangents of: 4 22 2y x x
34 4dy
x xdx
Horizontal tangents occur when slope = zero.34 4 0x x
3 0x x
2 1 0x x
1 1 0x x x
0, 1, 1x
Plugging the x values into the original equation, we get:
2, 1, 1y y y
(The function is even, so we only get two horizontal tangents.)
4 22 2y x x
4 22 2y x x
2y
4 22 2y x x
2y
1y
4 22 2y x x
4 22 2y x x
First derivative (slope) is zero at:
0, 1, 1x
34 4dy
x xdx