1
Q
Logarithmic Differentiation
A
- Understand that the Power Rule of Differentiation cannot be applied when the exponent is a variable.
- Take the natural log of both sides of an equation to simplify the problem by transforming a variable exponent into a product (of the variable with the natural log function).
- Apply logarithmic differentiation to complicated functions to make it easier to find derivatives by transforming exponents to products.
2
Q
note
A
- The Power Rule of Differentiation, d/dx(x n ) = nx^n − 1 , may only be applied when the exponent, n, is a fixed number. If there is a variable in the exponent, as with the function y = x^x , the best approach is to turn the exponent into a product
by taking the natural log of both sides of the equation. - To find the derivative, differentiate the resulting equation implicitly with respect to x.
- The expression (1/y)·y ́ will always be on the left side of the equation. To solve for y ́, multiply both sides of the equation by y.
- Then on the right side of the equation, replace y with its equivalent expression in terms of the x that was given at the very beginning of the problem.
- Apply logarithmic differentiation whenever the function
whose derivative is sought has a variable appearing in an exponent. - In this example the steps are laid out carefully.
- First, take the natural log of both sides and pull out cos x in the exponent so that it becomes a term in the product.
- Second, implicitly differentiate both sides, remembering to apply the Product Rule of Differentiation and the Chain Rule as necessary.
- Third, multiply both sides by y and replace y with (sin x) cos x .
3
Q
Differentiate the given function.
f(x)=x^x+1
A
f′(x)=x^x(xlnx+x+1)
4
Q
Differentiate the given function.
f(x)=(2x+3)^5x−7
A
f′(x)=(2x+3)^5x−8 ((10x+15)ln(2x+3)+10x−14)
5
Q
Differentiate the given function.
f(x)=(sinx)x^3x
A
f′(x)=(sinx)x^3x (cotx+3lnx+3)`
6
Q
Differentiate the given function.
f(x)=x^tanx
A
f′(x)=x^(tanx)−1(x(sec2x)lnx+tanx)
7
Q
Differentiate the given function.
f (x) = x^2x
A
f ′( x) = 2x^2x (ln x + 1)
8
Q
Differentiate the given function.
f (x) = (cos x)^x
A
f ′( x) = (cos x)^x (ln (cos x) − x tan x)
9
Q
Differentiate the given function.
f(x)=(cosx)^sinx
A
f′(x)=(cosx)^sinx ((cosx)ln(cosx)−sinxtanx)
10
Q
Differentiate the given function.
f(x)=(sinx)^x
A
f′(x)=(sinx)^x (ln(sinx)+xcotx)
11
Q
Differentiate the given function.
f(x)=(√7x−1)x^2
A
f′(x)=(√7x−1)^x^2 ((14x^2−2x)ln(7x−1)+7x^2/14x−2)
12
Q
Differentiate the given function.
f(x)=x(tanx)^x
A
f′(x)=(tanx)^x(1+xln(tanx)+x^2cscxsecx)
13
Q
Differentiate the given function.
f(x)=x^√2x+1
A
f′(x)=x^√2x+1 (xlnx+2x+1/x√2x+1)
14
Q
Differentiate the given function.
f(x)=(lnx)^lnx
A
f′(x)=(lnx)^lnx/x (ln(lnx)+1)
AP Calculus AB
flashcards
Decks in class (190)
# Cards
-
1.1.1 An Introduction to Thinkwell Calculus1
-
1.1.2 The Two Questions of Calculus3
-
1.1.3 Average Rates of Change14
-
1.1.4 How to Do Math1
-
1.2.1 Functions16
-
1.2.2 Graphing Lines14
-
1.2.3 Parabolas15
-
1.2.4 Some Non-Euclidean Geometry1
-
Chapter 1 Practice Test20
-
Chapter 1 Test20
-
2.1.1 Finding Rate of Change over an Interval18
-
2.1.2 Finding Limits Graphically9
-
2.1.3 The Formal Definition of a Limit6
-
2.1.4 The Limit Laws, Part I7
-
2.1.5 The Limit Laws, Part II17
-
2.1.6 One-Sided Limits11
-
2.1.7 The Squeeze Theorem13
-
2.1.8 Continuity and Discontinuity13
-
2.2.1 Evaluating Limits17
-
2.2.2 Limits and Indeterminate Forms12
-
2.2.3 Two Techniques for Evaluating Limits7
-
2.2.4 An Overview of Limits11
-
Chapter 2 Practice Test20
-
Chapter 2 Test20
-
3.1.1 Rates of Change, Secants, and Tangents12
-
3.1.2 Finding Instantaneous Velocity14
-
3.1.3 The Derivative13
-
3.1.4 Differentiability9
-
3.2.1 The Slope of a Tangent Line9
-
3.2.2 Instantaneous Rate14
-
3.2.3 The Equation of a Tangent Line17
-
3.2.4 More on Instantaneous Rate8
-
3.3.1 The Derivative of the Reciprocal Function19
-
3.3.2 The Derivative of the Square Root Function16
-
Chapter 3 Practice Test20
-
Chapter 3 Test20
-
4.1.1 A Shortcut for Finding Derivatives14
-
4.1.2 A Quick Proof of the Power Rule12
-
4.1.3 Uses of the Power Rule24
-
4.2.1 The Product Rule12
-
4.2.2 The Quotient Rule9
-
4.3.1 An Introduction to the Chain Rule15
-
4.3.2 Using the Chain Rule14
-
4.3.3 Combining Computational Techniques14
-
Chapter 4 Practice Test20
-
Chapter 4 Test20
-
5.1.1 A Review of Trigonometry13
-
5.1.2 Graphing Trigonometric Functions10
-
5.1.3 The Derivatives of Trigonometric Functions17
-
5.1.4 The Number Pi16
-
5.2.1 Graphing Exponential Functions13
-
5.2.2 Derivatives of Exponential Functions8
-
5.3.1 Evaluating Logarithmic Functions11
-
5.3.2 The Derivative of the Natural Log Function10
-
5.3.3 Using the Derivative Rules with Transcendental Functions8
-
Chapter 5 Practice Test20
-
Chapter 5 Test20
-
6.1.1 An Introduction to Implicit Differentiation13
-
6.1.2 Finding the Derivative Implicitly14
-
6.2.1 Using Implicit Differentiation14
-
6.2.2 Applying Implicit Differentiation9
-
6.3.1 The Exponential and Natural Log Functions12
-
6.3.2 Differentiating Logarithmic Functions12
-
6.3.3 Logarithmic Differentiation14
-
6.3.4 The Basics of Inverse Functions12
-
6.3.5 Finding the Inverse of a Function12
-
6.4.1 Derivatives of Inverse Function12
-
6.5.1 The Inverse Sine, Cosine, and Tangent Functions12
-
6.5.2 The Inverse Secant, Cosecant, and Cotangent Functions12
-
6.5.3 Evaluating Inverse Trigonometric Functions12
-
6.6.1 Derivatives of Inverse Trigonometric Functions12
-
6.7.1 Defining the Hyperbolic Functions12
-
6.7.2 Hyperbolic Identities12
-
6.7.3 Derivatives of Hyperbolic Functions12
-
Chapter 6 Practice Test20
-
Chapter 6 Test20
-
Practice Midterm Exam20
-
Midterm Exam20
-
7.1.1 Acceleration and the Derivative9
-
7.1.2 Solving Word Problems Involving Distance and Velocity15
-
7.2.1 Higher-Order Derivatives and Linear Approximation14
-
7.2.2 Using the Tangent Line Approximation Formula14
-
7.2.3 Newton's Method6
-
7.3.1 The Connection Between Slope and Optimization12
-
7.3.2 The Fence Problem8
-
7.3.3 The Box Problem8
-
7.3.4 The Can Problem8
-
7.3.5 The Wire-Cutting Problem5
-
7.4.1 The Pebble Problem8
-
7.4.2 The Ladder Problem6
-
7.4.3 The Baseball Problem7
-
7.4.4 The Blimp Problem7
-
Chapter 7 Practice Test20
-
Chapter 7 Test20
-
8.1.1 An Introduction to Curve Sketching7
-
8.1.2 Three Big Theorems12
-
8.2.1 Critical Points5
-
8.2.2 Maximum and Minimum13
-
8.2.3 Regions Where a Function Increases or Decreases12
-
8.2.4 The First Derivative Test5
-
8.3.1 Concavity and Inflection Points9
-
8.3.2 Using the Second Derivative to Examine Concavity6
-
8.4.1 Graphs of Polynomial Functions9
-
8.4.2 Cusp Points and the Derivative8
-
8.4.3 Domain-Restricted Functions and the Derivative6
-
8.4.4 The Second Derivative Test8
-
8.5.1 Vertical Asymptotes7
-
8.5.2 Horizontal Asymptotes and Infinite Limits9
-
8.5.3 Graphing Functions with Asymptotes5
-
8.5.4 Functions with Asymptotes and Holes5
-
8.5.5 Functions with Asymptotes and Critical Points5
-
Chapter 8 Practice Test20
-
Chapter 8 Test20
-
9.1.1 Antidifferentiation8
-
9.1.2 Antiderivatives of Powers of x9
-
9.1.3 Antiderivatives of Trigonometric and Exponential Functions11
-
9.2.1 Undoing the Chain Rule9
-
9.2.2 Integrating Polynomials by Substitution12
-
9.3.1 Integrating Composite Trigonometric Functions by Substitution12
-
9.3.2 Integrating Composite Exponential and Rational Functions by Substitution10
-
9.3.3 More Integrating Tirgonometric Functions by Substitution12
-
9.3.4 Choosing Effective Function Decompositions6
-
9.4.1 Approximating Areas of Plane Regions5
-
9.4.2 Areas, Riemann Sums, and Definite Integrals12
-
9.4.3 The Fundamental Theorem of Calculus, Part I12
-
9.4.4 The Fundamental Theorem of Calculus, Part II5
-
9.4.5 Illustrating the Fundamental Theorem of Calculus10
-
9.4.6 Evaluating Definite Integrals13
-
9.5.1 An Overview of Trigonometric Substitution Strategy12
-
9.5.2 Trigonometric Substitution Involving a Definite Integral: Part One2
-
9.5.3 Trigonometric Substitution Involving a Definite Integral: Part Two9
-
9.6.1 Deriving the Trapezoidal Rule12
-
9.6.2 An Example of the Trapezoidal Rule14
-
Chapter 9 Practice Test20
-
Chapter 9 Test20
-
10.1.1 Antiderivatives and Motion12
-
10.1.2 Gravity and Vertical Motion11
-
10.1.3 Solving Vertical Motion Problems8
-
10.2.1 The Area between Two Curves7
-
10.2.2 Limits of Integration and Area7
-
10.2.3 Common Mistakes to Avoid When Finding Areas6
-
10.2.4 Regions Bound by Several Curves6
-
10.3.1 Finding Areas by Integrating with Respect to y: Part One8
-
10.3.2 Finding Areas by Integrating with Respect to y: Part Two8
-
10.3.3 Area, Integration by Substitution, and Trigonometry8
-
10.4.1 Finding the Average Value of a Function8
-
10.5.1 Finding Volumes Using Cross-Sectional Slices8
-
10.5.2 An Example of Finding Cross-Sectional Volumes8
-
10.6.1 Solids of Revolution7
-
10.6.2 The Disk Method along the y-Axis9
-
10.6.3 A Transcendental Example of the Disk Method8
-
10.6.4 The Washer Method across the x-Axis8
-
10.6.5 The Washer Method across the y-Axis8
-
10.7.1 Introducing the Shell Method8
-
10.7.2 Why Shells Can Be Better Than Washers6
-
10.7.3 The Shell Method: Integrating with Respect to y6
-
10.8.1 An Introduction to Work12
-
10.8.2 Calculating Work12
-
10.8.3 Hooke's Law12
-
10.9.1 Center of Mass9
-
10.9.2 The Center of Mass of a Thin Plate11
-
10.10.1 An Introduction to Arc Length12
-
10.10.2 Finding Arc Lengths of Curves Given by Functions12
-
Chapter 10 Practice Test25
-
Chapter 10 Test25
-
11.1.1 An Introduction to Differential Equations12
-
11.1.2 Solving Separable Differential Equations13
-
11.1.3 Finding a Particular Solution12
-
11.1.4 Direction Fields15
-
11.1.5 Euler's Method for Solving Differential Equations Numerically13
-
11.2.1 Exponential Growth11
-
11.2.2 Logistic Growth13
-
11.2.3 Radioactive Decay10
-
Chapter 11 Practice Test12
-
Chapter 11 Test12
-
12.1.1 Indeterminate Forms14
-
12.1.2 An Introduction to L'Hôpital's Rule14
-
12.1.3 Basic Uses of L'Hôpital's Rule10
-
12.1.4 More Exotic Examples of Indeterminate Forms14
-
12.2.1 L'Hôpital's Rule and Indeterminate Products10
-
12.2.2 L'Hôpital's Rule and Indeterminate Differences14
-
12.2.3 L'Hôpital's Rule and One to the Infinite Power7
-
12.2.4 Another Example of One to the Infinite Power10
-
12.3.1 The First Type of Improper Integral10
-
12.3.2 The Second Type of Improper Integral10
-
12.3.3 Infinite Limits of Integration, Convergence, and Divergence11
-
Chapter 12 Practice Test15
-
Chapter 12 Test15
-
Practice Final Exam30
-
Final Exam30
