Calculus Lecture 5 Derivatives By Nur Uddin Ph

Calculus Lecture 5: Derivatives By: Nur Uddin, Ph. D 1

Overview • We discussed how to determine the slope of a curve at a point and how to measure the rate at which a function changes. • We have studied limits, we can define these ideas precisely and see that both are interpretations of the derivative of a function at a point. • We then extend this concept from a single point to the derivative function, and we develop rules for finding this derivative function easily, without having to calculate any limits directly. • The derivative is one of the key ideas in calculus, and is used to study a wide range of problems in mathematics, science, economics, and medicine. 2

Tangents and the Derivative at a Point 3

Example 1 4

Rate of change: derivative at a point 5

Interpretations for the limit of the difference quotient 6

The derivative as a function Remember the different to: 7

Example 8

Alternative formula for the derivative 9

Example 10

When does a function not have a derivative at a point? 11

When does a function not have a derivative at a point? (cont’d) 12

Differentiable functions are continuous 13

Differentiation Rules (1) 14

Differentiation Rules (2) 15

Differentiation Rules (2 a) 16

Differentiation Rules (3) 17

Differentiation Rules (4) 18

Example 19

Differentiation Rules (5) 20

Differentiation Rules (6) 21

Example 22

Differentiation Rules (7) 23

Second- and higher- order derivatives 24

Symbols for derivatives 25