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2.5 Implicit Differentiation Niagara Falls, NY & Canada Photo by Vickie Kelly, 2003
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.
3.7 Implicit Differentiation Implicitly Defined Functions –How do we find the slope when we cannot conveniently solve the equation to find the functions?
2.3 Rules for Differentiation Colorado National Monument Vista High, AB Calculus. Book Larson, V9 2010Photo by Vickie Kelly, 2003.
3.3 Differentiation Rules Colorado National Monument Photo by Vickie Kelly, 2003 Created by Greg Kelly, Hanford High School, Richland, Washington Revised.
Warm up Problems After correcting the homework, we will be taking Derivative Quiz #2.
6.2 Integration by Substitution M.L.King Jr. Birthplace, Atlanta, GA Greg Kelly Hanford High School Richland, Washington Photo by Vickie Kelly, 2002.
UNIT 2 LESSON 5 QUOTIENT RULE 1. 2 If you thought the product rule was bad...
Product & Quotient Rules Higher Order Derivatives Lesson 2.3.
Derivative Implicitly Date:12/1/07 Long Zhao Teacher:Ms.Delacruz.
Differentiation Revision for IB SL. Type of function Rule used to differentiate Polynomial Constant Always becomes zero Remember that, e, ln(3), are still.
2.4 Rates of Change and Tangent Lines Devils Tower, Wyoming Greg Kelly, Hanford High School, Richland, WashingtonPhoto by Vickie Kelly, 1993.
6.4 day 1 Separable Differential Equations Jefferson Memorial, Washington DC Greg Kelly, Hanford High School, Richland, WashingtonPhoto by Vickie Kelly,
Page 49 Introduction to Implicit Differentiation Q: What is the difference between f(x) = 2x 3 + 4x – 1 and y = 2x 3 + 4x – 1? A: Essentially none. If.
Section 3.3a. The Do Now Find the derivative of Does this make sense graphically???
8-2: Solving Systems of Equations using Substitution.
3.9 Derivatives of Exponential and Logarithmic Functions.
Unit 6 – Fundamentals of Calculus Section 6.4 – The Slope of a Curve No Calculator.
L’Hôpital’s Rule. L’Hopital’s Rule Analytically evaluate the following limit: By direct substitution you obtain the indeterminate form of type 0/0. 0/0,
Goldstein/Schneider/Lay/Asmar, CALCULUS AND ITS APPLICATIONS, 11e – Slide 1 of 33 Chapter 3 Techniques of Differentiation.
X and Y intercepts. X-intercept: The point where a line crosses the x-axis. The coordinates are ( x, 0) where x is any number on the x-axis Y-intercept:
2.4 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.
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.
This theorem allows calculations of area using anti-derivatives. What is The Fundamental Theorem of Calculus?
Solving Systems by Graphing or Substitution. Objective: To solve a system of linear equations in two variables by graphing or by substitution.
3.3 Rules for Differentiation AKA “Shortcuts”. Review from places derivatives do not exist: ▫Corner ▫Cusp ▫Vertical tangent (where derivative is.
3.1 Derivatives. Derivative A derivative of a function is the instantaneous rate of change of the function at any point in its domain. We say this is.
Warm Up 1)Sketch the graph of y = ln x a)What is the domain and range? b)Determine the concavity of the graph. c)Determine the intervals where the graph.
Parallel Lines. We have seen that parallel lines have the same slope.
2.6 The Derivative By Dr. Julia Arnold using Tan’s 5th edition Applied Calculus for the managerial, life, and social sciences text.
Differentiation – Product, Quotient and Chain Rules Department of Mathematics University of Leicester.
5.5 Differentiation of Logarithmic Functions By Dr. Julia Arnold and Ms. Karen Overman using Tan’s 5th edition Applied Calculus for the managerial, life,
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