Basic Differentiation Rules

Derivative Rules

• Theorem. [The Constant Rule] If k is a real number such that for all x in some open interval I, then for all

• Theorem. [The Power Rule] Let r be a rational number, and let . Then

for all values of x where this expression is defined.

f x k ' 0f x .x I

rf x x

1' rf x rx

Examples

• Find derivatives for the following functions:

• Find the equation of the line tangent to the graph of at the point 3y x 2,8 .

100f x x 13g x 5h x x

More Derivative Rules

• Theorem [The Constant Multiple Rule] Let k represent a real number, and let f be a differentiable function. Then the function kf is also differentiable and

• Example. Find the derivative of

' .d

kf x kf xdx

36 .f x x

• Theorem [The Sum and Difference Rules] Let f and g be differentiable functions. Then

and

• Example. Find the derivative of each function.

• Note. This theorem generalizes to any finite sum or difference.

' 'd

f x g x f x g xdx

' ' .d

f x g x f x g xdx

36 4f x x x 3 27 8 7 2g x x x x

Theorem.

Example. Find all values of x where the line tangent to the graph of has slope –1.

sin cosd

x xdx

cos sind

x xdx

siny x

The Derivative As a Rate of Change

• Slope.

lim

rise

run rate of change in with respect to

x c

f x f cdy

dx x cf x f c

x cy

x

y x

• Velocity. Let be a function giving the position of a point moving on a number line at time t.

s t

0

' lim

distance

time rate of change in position

t

s t s cs c

t cs t s c

t c

The derivative gives the instantaneous velocity at time 's c .t c

lim

instantaneous rate of change in with respect to

x c

f x f cdy

dx x cy x

The Derivative an Instantaneous Rate of Change

Example. A stone dropped from a bridge falls in t seconds. Find the velocity after 3 seconds. If a river flows 256 feet below the bridge, with what velocity does the rock enter the water?

216t