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.
2.3 Basic Differentiation Formulas
1n ndx nx
dx
Examples: 4f x x 34f x x 8y x 78y x
Power Rule:If n is any real number, then
Constant Multiple Rule:
d ducu c
dx dx
If c is a constant and f is differentiable function, then
1n ndcx cnx
dx
5 4 47 7 5 35dx x x
dx
Examples:
d du dvu v
dx dx dx
d du dvu v
dx dx dx
4 12y x x 34 12y x
4 22 2y x x 34 4dy
x xdx
The Sum Rule:
The Difference Rule:
Example:
Example:
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
2
0
2
Consider the function siny
We could make a graph of the slope: slope
1
0
1
0
1Now we connect the dots!
The resulting curve is a cosine curve.
sin cosd
x xdx
2
0
2
We can do the same thing for cosy slope
0
1
0
1
0The resulting curve is a sine curve that has been reflected about the x-axis.
cos sind
x xdx
d dv duuv u v
dx dx dx Notice that this is not just the
product of two derivatives.
This is sometimes memorized as: d uv u dv v du
2 33 2 5d
x x xdx
2 3x 26 5x 32 5x x 2x
4 2 2 4 26 5 18 15 4 10x x x x x
4 210 33 15x x
The Product 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
The Quotient Rule:
Example:
We can find the derivative of the tangent function by using the quotient rule.
tand
xdx
sin
cos
d x
dx x
2
cos cos sin sin
cos
x x x x
x
2 2
2
cos sin
cos
x x
x
2
1
cos x
2sec x
2tan secd
x xdx