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.
Consider the function y = 4 graphedto the right. Clearly, the slope of all the tangent lines onthis graph would all be 0.
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
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 x values of 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
product rule:
Notice that this is not just the product of two derivatives.
dx
dvuv
dx
duuv
dx
d)(
)52)(3()( 32 xxxxh
Since multiplication is commutative, the product rule order can change. The book uses this one.
I do not.
another way to remembered it :
467)( 3 xxxh
f(x) g(x)
so )()()()()( xgxfxgxfxh
der. of 1st times the 2nd + 1st times the der. of the 2nd
68421
167421)(23
32
xx
xxxxh
d dv duuv u v
dx dx dx
quotient rule:
2
du dvv ud u dx dx
dx v v
or 2
u v du u dvdv v
3
2
2 5
3
d x x
dx x
2 2 3
22
3 6 5 2 5 2
3
x x x x x
x
Since it is subtraction, the order does matter
2)(
)()()()()(
xg
xgxfxgxfxh
4
62
)(
)()(
2
4
x
xx
xg
xfxh
Same as product, but subtraction in thenumerator
22
423
4
)2)(62()4)(68()(
x
xxxxxxh
What is the velocity as a function of time (in seconds) of the time-position function meters?
5
3 4)(
t
ttts
Example: working with numerical values
Let y = uv be the product of the functions u and v. Find y’(2) if
2)2(
,1)2(
,4)2(
,3)2(
v
v
u
u
Ans: Find the product
y’ = u’ v + u v’ y’(2)= (-4)(1)+(3)(2) = -4+6 = 2
Higher Order Derivatives:
dyy
dx is the first derivative of y with respect to x.
2
2
dy d dy d yy
dx dx dx dx
is the second derivative.
(y double prime)
dyy
dx
is the third derivative.
4 dy y
dx is the fourth derivative.
We will learn later what these higher order derivatives are used for.