If the derivative of a function is its slope, then for a
constant function, the derivative must be zero.
( )
0
dc
dx =
example:
3y=
0y′=
The derivative of a constant is zero.
2.3 Basic Differentiation
Formulas
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Differentiation Formulas - Calculus I - Lecture Slides, Slides of Calculus
In my class of Calculus-I, I take lecture note from these slides, hope these lecture slides help other student.The key point in these slides are:Differentiation Formulas, Derivative of Function, Derivative of Constant, Constant Function, Power Rule, Real Number, Constant Multiple Rule, Differentiable Function, Sum Rule, Difference Rule, Horizontal Tangents
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If the derivative of a function is its slope, then for a
constant function, the derivative must be zero.
d c dx
example: y = 3 y′ = 0
The derivative of a constant is zero.
2.3 Basic Differentiation
Formulas
( )
d (^) n n 1 x nx dx
−
Examples: f ( x ) = x^4 f ′ ( x ) = 4 x^3
8 y = x
7 y ′ = 8 x
Power Rule:
If n is any real number, then
Constant Multiple Rule:
( )
d du cu c dx dx
If c is a constant and f is differentiable function, then
d n n 1 cx cnx dx
−
5 4 4 7 7 5 35
d x x x dx
Examples:
Example:
Find the horizontal tangents of:
4 2
y = x − 2 x + 2
3
dy
x x
dx
Horizontal tangents occur when slope = zero.
3
4 x − 4 x = 0
3
x − x = 0
( )
2
x x − 1 = 0
x x ( + (^1) )( x− (^1) ) = 0
x = 0, −1, 1
Plugging the x values into the original equation, we get:
y = 2, y = 1, y = 1
0
1
2
3
4
-2 -1 1 2
4 2
y = x − 2 x + 2
y = 2
y = 1
2
0
2
−
− π
We can do the same thing for y = cos(^ θ)
θ slope
0
1
0
− 1
0
The resulting curve is a sine curve that has
been reflected about the x-axis.
cos ( ) sin
d x x dx
= −
2.4 The Product and
Quotient Rules
2
du dv v u d u (^) dx dx
dx v v
or (^) 2
u v du u dv d v v
^ − =
3
2
2 5
3
d x x
dx x
( )( ) ( ) (^ )
( )
2 2 3
2 2
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.
tan
d x dx
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
cos x
2 sec x
2 tan sec
d x x dx
=