The Product and Quotient Rules

Slides:

Advertisements

Similar presentations

Consider the function We could make a graph of the slope: slope Now we connect the dots! The resulting curve is a cosine curve. 2.3 Derivatives of Trigonometric.

Advertisements

Complex Functions These derivatives involve embedded composite, product, and quotient rules. The functions f or g must be derived using another rule.

Maximum ??? Minimum??? How can we tell?

Engineering Mathematics: Differential Calculus. Contents Concepts of Limits and Continuity Derivatives of functions Differentiation rules and Higher Derivatives.

Remember: Derivative=Slope of the Tangent Line.

Basic Skills in Higher Mathematics Robert Glen Adviser in Mathematics Mathematics 1(H) Outcome 3.

More On The Derivative Rules

Higher Mathematics: Unit 1.3 Introduction to Differentiation

Limits Pre-Calculus Calculus.

Sec. 2.1: The Derivative and the Tangent Line

Do Now Find the tangents to the curve at the points where

THE DISTANCE FORMULA During this lesson, we will use the Distance Formula to measure distances on the coordinate plane.

Every slope is a derivative. Velocity = slope of the tangent line to a position vs. time graph Acceleration = slope of the velocity vs. time graph How.

Differentiation Recap The first derivative gives the ratio for which f(x) changes w.r.t. change in the x value.

SEC. 4.3 USING DERIVATIVES FOR CURVE SKETCHING. IN THE LEFT HAND COLUMN ARE GRAPHS OF SEVERAL FUNCTIONS. IN THE RIGHT- HAND COLUMN – IN A DIFFERENT ORDER.

AP Calculus BC September 9, 2015 Day 7 – The Chain Rule and Implicit Differentiation.

Power of a Product and Power of a Quotient Let a and b represent real numbers and m represent a positive integer. Power of a Product Property Power of.

3.3 Rules for Differentiation What you’ll learn about Positive integer powers, multiples, sums, and differences Products and Quotients Negative Integer.

Differentiation Formulas

Derivatives By Mendy Kahan & Jared Friedman. What is a Derivative? Let ’ s say we were given some function called “ f ” and the derivative of that function.

Dr. Omar Al Jadaan Assistant Professor – Computer Science & Mathematics General Education Department Mathematics.

Chapter 3.1 Tangents and the Derivative at a Point.

Definition of Derivative.  Definition   f‘(x): “f prime of x”  y‘ : “y prime” (what is a weakness of this notation?)  dy/dx : “dy dx” or, “the derivative.

Examples A: Derivatives Involving Algebraic Functions.

Using Derivatives for Curve Sketching

13.1 Antiderivatives and Indefinite Integrals. The Antiderivative The reverse operation of finding a derivative is called the antiderivative. A function.

Consider the function We could make a graph of the slope: slope Now we connect the dots! The resulting curve is a cosine curve.

EXAMPLE 1 Find a positive slope Let (x 1, y 1 ) = (–4, 2) = (x 2, y 2 ) = (2, 6). m = y 2 – y 1 x 2 – x 1 6 – 2 2 – (–4) = = = Simplify. Substitute.

Warm Up Write an equation of the tangent line to the graph of y = 2sinx at the point where x = π/3.

Derivatives of Trigonometric Functions. 2 Derivative Definitions We can now use the limit of the difference quotient and the sum/difference formulas.

Exercise 7K Page 139 Questions 4, 5, 7, 12 & 13. Exercise 7K Page 139 Q4 (a) Factorise f(x) = x 3 + x 2 – 16x – – 16 – 16 Try ± 1, ± 2, ±

In the past, one of the important uses of derivatives was as an aid in curve sketching. Even though we usually use a calculator or computer to draw complicated.

Logarithmic Functions. Examples Properties Examples.

Derivative Shortcuts -Power Rule -Product Rule -Quotient Rule -Chain Rule.

Finding the Slope. The Question Given the two point (-3,1) and (3,3) which lie on a line, determine the slope of that line. First, let’s draw the graph,

Chapter 16B.5 Graphing Derivatives The derivative is the slope of the original function. The derivative is defined at the end points of a function on.

Slope Formula. P1: ( -4, -5) P1: ( x 1, y 1 ) X 1 = -4, y 1 = -5 P2:( 4, 2) P2: (x 2, y 2 ) X 2 = 4, y 2 = 2 ( -4, -5) & ( 4,2)

2.3 Basic Differentiation Formulas

Ch. 3 – Derivatives 3.3 – Rules for Differentiation.

Calculus Section 3.1 Calculate the derivative of a function using the limit definition Recall: The slope of a line is given by the formula m = y 2 – y.

A Brief Introduction to Differential Calculus. Recall that the slope is defined as the change in Y divided by the change in X.

Basic derivation rules We will generally have to confront not only the functions presented above, but also combinations of these : multiples, sums, products,

Finding the Equation of a Line Using Point Slope Formula

3.3 – Rules for Differentiation

The Product and Quotient rules

Warm Up Determine for y2 + xy + 3x = 9.

Ch. 11 – Limits and an Introduction to Calculus

Implicit Differentiation

Quadratic Equations Chapter 5.

The Quotient Rule The Quotient Rule is used to find derivatives for functions written as a fraction:

Derivatives of Trig Functions

2.3 Basic Differentiation Formulas

Derivatives of Trig Functions

Derivatives of Trig Functions

Using Derivatives For Curve Sketching

Double Chain Rule.

Second Derivative 6A.

5.3 Using Derivatives for Curve Sketching

Stationary Point Notes

Section 2.3 Day 1 Product & Quotient Rules & Higher order Derivatives

Differentiation Recap

Differentiation Summary

Lesson 4-3: Using Derivatives to Sketch Curves

PARTIAL DIFFERENTIATION 2

The Chain Rule Section 3.6b.

CALCULUS I Chapter II Differentiation Mr. Saâd BELKOUCH.

A Brief Introduction to Differential Calculus

Quadratic Functions Chapter 5.

4.3 Using Derivatives for Curve Sketching.

Presentation transcript:

The Product and Quotient Rules

If you recall, our formula for slope is defined as: we can also write this as: Therefore: Therefore: can be written as:

Let: Then: You can see this relationship by looking at the following diagram:

So:

Therefore, if then: This is called the Product Rule

Find the derivative of: Answer:

Find the derivative of: Answer:

Find the derivative of: Answer: Find the derivative of: Answer:

Let: Then: If: Then:

This is called the Quotient Rule

Find the derivative of: Answer: Find the derivative of: Answer:

Find the derivative of: Answer: Find the derivative of: Answer:

Find the derivative of: Answer: Find the derivative of: Answer: Find the derivative of: Answer: Find the derivative of: Answer:

Find the derivative of: Answer: Find the derivative of: Answer:

Find the coordinates of any stationary points on the following curves and state the nature of each: Answer: Max at (1, 256), Min at (5, 0)