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

4 22 2y x x

2y

1y

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

2.4 The Product and Quotient Rules

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

Derivatives of the remaining trigonometric functions can be determined the same way.

sin cosd

x xdx

cos sind

x xdx

2tan secd

x xdx

2cot cscd

x xdx

sec sec tand

x x xdx

csc csc cotd

x x xdx