GOAL: USE DEFINITION OF DERIVATIVE TO FIND SLOPE, RATE OF CHANGE,

INSTANTANEOUS VELOCITY AT A POINT.

3.1 Definition of Derivative

Definition:

The derivative of f(x) with respect to x is the function   and is defined as,

This is read “f prime of x”

Find the derivative:

Hint: use (x+h) and (x) into definition

2. Use definition of derivative to find the slope, rate of change, and velocity:

Using the last equation for the derivative, find the slope of the tangent at x=2

Rate of change at x= -3Instantaneous velocity at x= 9

Find the derivative by using the definition of derivative:

3. 𝑓 (𝑡 )= 𝑡𝑡+1

𝑓 ′ (1 ) , 𝑓 ′ (0 ) , 𝑓 ′ (−2 )

Find the derivative by rationalization:

4. 𝑓 (𝑧 )=√5 𝑧+8

Find the derivative by finding left and right limits

5. 𝐹𝑖𝑛𝑑 𝑓 ′ (0 ) h𝑤 𝑒𝑛 𝑓 (𝑥 )=¿𝑥∨¿

Differentiability

Definition

A function is called differentiable at f(x) at x=a if exists and is called differentiableon an interval if the derivative exists for each point in the interval.

Theorem

If f(x) is differentiable at x=a, then f(x) is continuous at x=a.

Different notations referring to the derivative of f(x) with respect to x

𝑓 ′ (𝑥 )𝑦 ′

𝑑𝑦𝑑𝑥

𝑑𝑑𝑥

𝑓 (𝑥 )

𝑑𝑓𝑑𝑥𝑑𝑑𝑥

(𝑦 )

'

'

x a

x a

x a

x a

x a

x a

f x

y

dy

dx

df x

dxdf

dx

dy

dx

6. Determine the graph of the derivative

Tangent Line and Normal Line

Slope of the tangent line is The normal line is perpendicular to the

tangent line. The slope is the opposite reciprocal of the tangent line.

Find the tangent line and normal line at x= -1

check graphs 3.1

Homework

3.1 InterActMath.com