Finding DerivativesSections 2.1, 2.2, 5.1, and 5.4

Definition• The derivative of a function tells us the instantaneous rate of

change of a function.

• We can see the derivative by looking at the average rate of change over a decreasing interval.

The definition of derivative

• The derivative of x2 + x would be given by

0

( ) ( )limh

f x h f x

h

2 2

0

( ) ( ) ( )limh

x h x h x x

h

Use the definition of definition of derivative to find the derivative of each of the following functions:

• x2 + 5x - 7 • 3x2 – 4x + 6

The real way to take a derivative• For a polynomial, we can take the derivative in two steps.• 1. Bring down the exponent and multiply it times the coefficient

of x.• 2. Subtract 1 from each exponent of x.

• Find the derivative of 3x2 – 5x + 6

Find the derivative for each of the following• x2 + 6x – 7 • 3x2 + 5x + 2

More Derivatives• Before you try to take the derivative, make sure that

everything is converted to a rational exponent.

2 15xx

32 6x x x

ex and ln x• The derivative of ex is ex

• The derivative of ln x is 1/x

• Find the derivative of 3x2 + 5x – x-1/2 + ln x

Finding slope• Since the derivative tells the rate of change of a function, we

can substitute a specific x-value into the derivative to find the slope of the tangent line to the curve at a point.• Find the slope of the tangent line to each curve at x = 4.

• x2 + 3x – 2

• 2x2 + 5 – 1/x

Equation of Tangent Line• Find the equation of a tangent line at x = 4 for each of the

following functions.• -3x2 + 17

• 2x2 + 5 –