Lesson 3-7: Implicit Differentiation AP Calculus Mrs. Mongold.

Slides:

Advertisements

Similar presentations

2.5 Implicit Differentiation Niagara Falls, NY & Canada Photo by Vickie Kelly, 2003.

Advertisements

However, some functions are defined implicitly. Some examples of implicit functions are: x 2 + y 2 = 25 x 3 + y 3 = 6xy.

DIFFERENTIATION & INTEGRATION CHAPTER 4.  Differentiation is the process of finding the derivative of a function.  Derivative of INTRODUCTION TO DIFFERENTIATION.

Implicit Differentiation

2.5 The Chain Rule If f and g are both differentiable and F is the composite function defined by F(x)=f(g(x)), then F is differentiable and F′ is given.

2.5 Implicit Differentiation. Implicit and Explicit Functions Explicit FunctionImplicit Function But what if you have a function like this…. To differentiate:

Implicit Differentiation. Objectives Students will be able to Calculate derivative of function defined implicitly. Determine the slope of the tangent.

Before we start, we are going to load the Calculus Tools flash application software to your calculator. 1. Connect the sending and.

3.6 Derivatives of Logarithmic Functions 1Section 3.6 Derivatives of Log Functions.

Implicit Differentiation

Implicit Differentiation

Implicit Differentiation 3.6. Implicit Differentiation So far, all the equations and functions we looked at were all stated explicitly in terms of one.

Quiz corrections due Friday. 2.5 Implicit Differentiation Niagara Falls, NY & Canada Greg Kelly, Hanford High School, Richland, WashingtonPhoto by Vickie.

1 Implicit Differentiation Lesson Introduction Consider an equation involving both x and y: This equation implicitly defines a function in x It.

Section 2.5 – Implicit Differentiation. Explicit Equations The functions that we have differentiated and handled so far can be described by expressing.

3.7 Implicit Differentiation Niagara Falls, NY & Canada Greg Kelly, Hanford High School, Richland, WashingtonPhoto by Vickie Kelly, 2003.

Implicit Differentiation Niagara Falls, NY & Canada Greg Kelly, Hanford High School, Richland, WashingtonPhoto by Vickie Kelly, 2003.

1 Implicit Differentiation. 2 Introduction Consider an equation involving both x and y: This equation implicitly defines a function in x It could be defined.

Calculus: IMPLICIT DIFFERENTIATION Section 4.5. Explicit vs. Implicit y is written explicitly as a function of x when y is isolated on one side of the.

Section 2.5 Implicit Differentiation

Ms. Battaglia AB/BC Calculus. Up to this point, most functions have been expressed in explicit form. Ex: y=3x 2 – 5 The variable y is explicitly written.

In this section, we will investigate a new technique for finding derivatives of curves that are not necessarily functions.

Lesson: Derivative Techniques - 4  Objective – Implicit Differentiation.

3.7 Implicit Differentiation Niagara Falls, NY & Canada Greg Kelly, Hanford High School, Richland, WashingtonPhoto by Vickie Kelly, 2003.

Slide 3- 1 Quick Quiz Sections 3.4 – Implicit Differentiation.

3.7 Implicit Differentiation Niagara Falls, NY & Canada Greg Kelly, Hanford High School, Richland, WashingtonPhoto by Vickie Kelly, 2003.

Implicit Differentiation Objective: To find derivatives of functions that we cannot solve for y.

Section 3.5 Implicit Differentiation 1. Example If f(x) = (x 7 + 3x 5 – 2x 2 ) 10, determine f ’(x). Now write the answer above only in terms of y if.

3.7 Implicit Differentiation Niagara Falls, NY & Canada Greg Kelly, Hanford High School, Richland, WashingtonPhoto by Vickie Kelly, 2003.

Logarithmic Differentiation

Calculus and Analytical Geometry

Lesson: ____ Section: 3.7  y is an “explicitly defined” function of x.  y is an “implicit” function of x  “The output is …”

3.8 Implicit Differentiation Niagara Falls, NY & Canada Photo by Vickie Kelly, 2003.

Implicit Differentiation 3.5. Explicit vs. Implicit Functions.

René Descartes 1596 – 1650 René Descartes 1596 – 1650 René Descartes was a French philosopher whose work, La géométrie, includes his application of algebra.

UNIT 2 LESSON 9 IMPLICIT DIFFERENTIATION 1. 2 So far, we have been differentiating expressions of the form y = f(x), where y is written explicitly in.

Implicit Differentiation Niagara Falls, NY & Canada Greg Kelly, Hanford High School, Richland, WashingtonPhoto by Vickie Kelly, 2003.

An explicit equation is one that is solved for y. An implicit equation is one where the y’s are sprinkled throughout the equation. All the same rules.

3.5 Implicit Differentiation

2.5 The Chain Rule If f and g are both differentiable and F is the composite function defined by F(x)=f(g(x)), then F is differentiable and F′ is given.

Implicit Differentiation

Implicit Differentiation

Implicit Differentiation

Implicit Differentiation

4.2 – Implicit Differentiation

3.6 Warm-Up Find y´´ Find the Derivative:.

4.2 Implicit Differentiation

Implicit Differentiation

MTH1170 Implicit Differentiation

Implicit Differentiation Implicit differentiation

Implicit Differentiation Continued

4.2 – Implicit Differentiation

3.7 Implicit Differentiation

Implicit Differentiation

Implicit Differentiation

Implicit Differentiation

Implicit Differentiation

Implicit Differentiation

3.6 Implicit Differentiation

Unit 3 Lesson 5: Implicit Differentiation

Implicit Differentiation

2.5 Implicit Differentiation

IMPLICIT Differentiation.

Implicit Differentiation & Related Rates

3.7 Implicit Differentiation

3.7 Implicit Differentiation

7. Implicit Differentiation

Presentation transcript:

Lesson 3-7: Implicit Differentiation AP Calculus Mrs. Mongold

Implicit Differentiation Use this when we have an equation we want to differentiate that cannot be solved for y. Differentiate using all rules to date with respect to x and whenever differentiating something with a y we have to put dy/dx behind it.

This is not a function, but it would still be nice to be able to find the slope. Do the same thing to both sides. Note use of chain rule.

This can’t be solved for y. This technique is called implicit differentiation. 1 Differentiate both sides w.r.t. x. 2 Solve for.

We need the slope. Since we can’t solve for y, we use implicit differentiation to solve for. Find the equations of the lines tangent and normal to the curve at. Note product rule.

Find the equations of the lines tangent and normal to the curve at. tangent:normal:

Higher Order Derivatives Find if. Substitute back into the equation.

Homework Page