Using the Quotient Rule Twice to Find the Second Derivative
Using the quotient rule twice to find the second derivative of a rational function
Finding a Derivative by Simplifying a Rational
Author: ShaNisaa RaSun
The following problem is solved in this video. It is recommended that you try to solve the problem before watching the video. You can click "Answer" to reveal the answer to the problem.
Problem: Find the derivative of the following function \(g(x)=\dfrac{35x^4-12x^3+2}{9x^2}.\)
\(g'(x) = \dfrac{70}{9}x - \dfrac{4}{3} - \dfrac{4}{9}x^{-3}\)
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Determining the properties of a function from the graph of its derivative
MLC WIR 20B M151 week9 #2
Using the derivative to determine where a function is increasing and decreasing
MLC WIR 20B M151 week9 #3
Finding the absolute maximum and minimum values of a function on a closed interval
MLC WIR 20B M151 week9 #4
Finding the intervals where a function is increasing and decreasing and its local maximums and minimums
MLC WIR 20B M151 week9 #5
Finding where a function is increasing, decreasing, concave up and concave down and its local extrema and inflection points
MLC WIR 20B M151 week9 #6
Finding where a function is increasing, decreasing, concave up and concave down and its local extrema and inflection points
MLC WIR 20B M151 week9 #7
Finding where a function is increasing, decreasing, concave up and concave down and its local extrema and inflection points
MLC WIR 20B M151 week9 #8a
Evaluating a limit using logarithm properties
MLC WIR 20B M151 week9 #8b
Evaluating a limit with an indeterminate difference using a common denominator
MLC WIR 20B M151 week9 #8c
Evaluating a limit using L'Hospital's Rule
MLC WIR 20B M151 week9 #8d
Evaluating a limit with an indeterminant power using L'Hospital's Rule
MLC WIR 20B M151 week9 #8e
Evaluating a limit with an indeterminant product using L'Hospital's Rule
MLC WIR 20B M151 week9 #9
Finding a number that satisfies the conclusion of the Mean Value Theorem for a function on an interval
MLC WIR 20B M151 week9 #8f
Evaluating a limit with an indeterminant product using L'Hospital's Rule
MLC WIR 20B M151 week9 #10a
Finding the antiderivative of a polynomial
MLC WIR 20B M151 week10 #6a
Evaluating a a limit from the left for a rational function
MLC WIR 20B M151 week10 #6b
Evaluating the limit of a rational function by factoring and cancelling
MLC WIR 20B M151 week10 #8a
Evaluating the limit at infinity of a rational function
MLC WIR 20B M151 week10 #9a
Finding a derivative using the Product Rule
MLC WIR 20B M151 week10 #9b
Finding a derivative using the Chain Rule multiple times
MLC WIR 20B M151 week10 #9c
Finding a derivative using the Chain Rule and Product Rule
MLC WIR 20B M151 week10 #9d
Finding a derivative using the Quotient Rule
MLC WIR 20B M151 week10 #10a
Finding the derivative of an implicit function
MLC WIR 20B M151 week10 #10b
Finding the derivative of an implicit function
MLC WIR 20B M151 week10 #11
Using logarithmic differentiation to find the derivative of a function
MLC WIR 20B M151 week10 #12
Finding the tangent line to a parametric curve
MLC WIR 20B M151 week10 #13
Finding when the tangent line to a parametric curve is vertical or horizontal
MLC WIR 20B M151 week10 #15
Determining the rate that the area of a triangle is increasing based on the rate an angle is increasing
MLC WIR 20B M151 week10 #16
Using a linear approximation to estimate the value of a function
MLC WIR 20B M151 week10 #17
Finding the value for a function that satisfies the Mean Value Theorem on an interval
MLC WIR 20B M151 week10 #18
Determining the properties of a function from the graph of its derivative
MLC WIR 20B M151 week10 #19a
Using L'Hospital's Rule to solve a limit
MLC WIR 20B M151 week10 #19b
Using L'Hospital's Rule to solve a limit with an indeterminant product
MLC WIR 20B M151 week10 #19c
Using L'Hospital's Rule to solve a limit with an indeterminant difference
MLC WIR 20B M151 week10 #19d
Using L'Hospital's Rule to solve a limit with an indeterminant power
MLC WIR 20B M151 week10 #24b
Evaluating a definite integral by first simplifying the integrand
Review for the Common Exam: MATH 152 Exam 1 a Review Problems 1-4
Reviewing u-Substitution
Review for the Common Exam: MATH 152 Exam 1 b Review Problem 36
Review of partial fractions
Review for the Common Exam: MATH 152 Exam 1 b Review Problems 34 & 35
Review of partial fractions
WIR 20B M152 V88
Finding the length of a curve given by parametric equations
WIR 20B M152 V90
Finding the surface area of a rotated parametric curve
WIR 20B M152 V81
Finding the Taylor series of a function
WIR 20B M152 V82
Finding the Taylor series of a function
WIR 20B M152 V70
Determining if a series is absolutely convergent, conditionally convergent, or divergent.
WIR 20B M152 V76
Finding a power series representation for a function
WIR 20B M152 V2
Integrating a rational function using u-substitution
WIR 20B M152 V3
Integrating a rational function using u-substitution
MATH 152: u-Substitution Exercise 3
Using u-substitution to integrate a rational function
Cases for Partial Fraction Decomposition
Explaining all four cases of partial fraction decomposition
MATH 152: Partial Fraction Decomposition Exercise 1
Using partial fraction decomposition to rewrite and then integrate a rational function
MATH 152: Partial Fraction Decomposition Exercise 2
Using partial fraction decomposition to rewrite and then integrate a rational function
MATH 152: Partial Fraction Decomposition Exercise 3
Using partial fraction decomposition to rewrite and then integrate a rational function
MATH 152: Partial Fraction Decomposition Exercise 5
Using partial fraction decomposition to rewrite and then integrate a rational function
MATH 152: Partial Fraction Decomposition Exercise 6
Using partial fraction decomposition to rewrite and then integrate a rational function
Vector Functions: MATH 171 Problems 1 & 2
Proving a certain parameterization gives the unit circle and showing vector equations of lines are parallel
Vectors and Derivatives (Product Rule): MATH 171 Problems 1-3
Proving facts about the derivatives of vector functions including the product rule
Vectors and Derivatives: MATH 151 K.1 Problems 1-8
Derivatives and vectors with some physics applications
Derivatives and Applications: MATH 151 K.2 Problems 9-15
Tangent lines to parametric equations and related rates examples
Derivatives and Applications: MATH 171 Problems 7 & 8
Calculating slopes of tangent lines to parametric curves
Review for the Common Exam: MATH 151 Exam 2 Review Problems 10-15
Implicit differentiation and physics applications of derivatives
Review for the Common Exam: MATH 151 Exam 2 Review Problems 16-20
Derivatives of parametric equations and tangent lines
Review for the Common Exam: MATH 151 Exam 3 Review Problems 1-7
Derivatives of exponential functions and the exponential model
Review for the Common Exam: MATH 151 Exam 2 Review Problems 1-9
Reviewing the chain rule and the derivatives and limits of trigonometric functions
Review for Common Exam: MATH 151 Exam 3 Review Problems 22-24
Optimization problems
Review for Common Exam: MATH 151 Exam 3 Review Problems 25-30
Antiderivatives and physics applications
Inverse Trigonometric Functions: MATH 171 Problems 4-6
Properties of inverse trig functions and the derivative of arctangent
Limits: MATH 151 Problems 6-20
Evaluating limits of functions
Limits: MATH 151 Problems 6-20
Evaluating Limits of Functions
Continuity: MATH 151 2.5 Problems 1-6
Continuity of Functions and the Intermediate Value Theorem
Continuity: MATH 171 Problems 6 & 7
Proving a piecewise function and a polynomial are continuous
Approximations and Exponentials: MATH 151 2.6 Problems 7-14
Approximation and Newton's Method, and limits and derivatives of exponential functions
Limits and Asymptotes: MATH 151 2.6 Problems 7-11
Limits at infinity and asymptotes, along with physics applications
Limits and Asymptotes: MATH 151 2.7 Problems 7-11
Limits at infinity, asymptotes, and tangent lines
Differentiation Formulas: MATH 171 Problems 6 & 7
Proving the Product Rule and the rule for the derivative of a difference of functions
Differentials: MATH 171 Problems 1 & 2
Proving facts about differentials including a product rule
Trigonometry Derivatives and the Chain Rule: MATH 171 Problems 1-3
Proving the derivatives of trigonometric functions and that sine is continuous
Trigonometry Derivatives and the Chain Rule: MATH 151 3.3 Problems 1-9
Derivatives of trigonometric functions and using the Chain Rule
Approximations and Exponentials: MATH 151 3.4 Problems 7-14
Approximation and Newton's Method, and limits and derivatives of exponential functions
Trigonometry Derivatives and the Chain Rule: MATH 151 3.4 Problems 1-9
Derivatives of trigonometric functions and using the Chain Rule
Vectors and Derivatives: MATH 151 3.4 Problems 1-8
Derivatives and vectors with some physics applications
Trigonometry Derivatives and the Chain Rule: MATH 171 Problems 4 & 5
Using the Chain Rule to prove facts about derivatives
Exponentials and Hyperbolic Trigonometric Functions: MATH 171 Problems 3 & 4
Examples involving the tangent line to an exponential function and finding the derivative of hyperbolic cosine
Implicit Differentiation: MATH 151 3.5 Problems 10-13
Implicit differentiation and finding tangent lines
Derivatives and Applications: MATH 151 3.9 Problems 9-15
Tangent lines to parametric equations and related rates examples
Absolute Extrema: MATH 171 Problems 4 & 5
Proving Fermat's Theorem and graphing local and absolute extrema
Mean Value Theorem: MATH 171 Problems 6-8
Using the Mean Value Theorem to prove facts about functions
Studying the Shape of Curves: MATH 171 Problems 1-3
What the derivatives of a function tell us about the shape of its graph
L'Hospital's Rule: MATH 171 Problems 7-9
Proving L'Hospital's Rule and using it to evaluate limits
Optimization: MATH 171 Problems 1-3
Optimizing values in word problems
Antiderivatives: MATH 171 Problems 4-6
Proving facts about antiderivatives and a physics application
Fundamental Theorem of Calculus: MATH 171 Problems 1-3
Proving and then applying the Fundamental Theorem of Calculus
Integration by Substitution: MATH 151 5.5 Problems 13 & 14
Examples of integration by substitution
Review for the Common Exam: MATH 152 Exam 1 a Review Problems 1-4
Reviewing u-Substitution
Review for the Common Exam: MATH 152 Exam 1 b Review Problem 36
Review of partial fractions
Review for the Common Exam: MATH 152 Exam 1 b Review Problems 34 & 35
Review of partial fractions
Increasing, Decreasing, and Non Monotonic Sequences
Explaining increasing, decreasing, and non monotonic for sequences
Integration by Parts: MATH 172 Problems 1 & 2
Proving the formula for integration by parts and a reduction formula
Integration Using Partial Fractions: MATH 172 Problems 8 & 9
Finding a partial fraction decomposition and integrating using partial fractions
Cases for Partial Fraction Decomposition
Explaining all four cases of partial fraction decomposition
WIR2 20B M251 V6
Determining the domain of a three-dimensional vector function
WIR2 20B M251 V7
Finding the limit of a three-dimensional vector function
WIR2 20B M251 V8
Finding a vector equation for the tangent line to a three-dimensional vector function
WIR2 20B M251 V9
Finding the angle of intersection for two three-dimensional vector functions
WIR2 20B M251 V10
Finding the length of a three-dimensional curve
WIR2 20B M251 V11
Finding the curvature of a three-dimensional vector function
WIR4 20B M251 V9
Finding the absolute maximum and minimum values of a function of two variables on a rectangle
WIR4 20B M251 V10
Finding the absolute maximum and minimum values of a function of two variables on a triangular region
WIR8 20B M251 V1
Evaluating a line integral along half a circle
WIR8 20B M251 V2
Evaluating a line integral along a curve in space
WIR8 20B M251 V3
Evaluating a line integral along a curve
WIR8 20B M251 V4
Evaluating a line integral along a curve
WIR8 20B M251 V5
Evaluating a line integral along a curve in space
WIR8 20B M251 V6
Evaluating a line integral of a vector field along an curve
WIR8 20B M251 V7
Evaluating a line integral of a vector field along a curve in space
WIR8 20B M251 V8
Using a line integral to find the work done by a force field moving a particle along a curve
Matrix Derivative and Integral
Explaining how to find the derivative and integral of matrix and vector functions
WIR 20B M152 V45
Determining if a sequence converges or diverges
MATH 152: Sequences Exercise 4
Finding the limit of a sequence
MATH 152: Sequences Exercise 6
Finding the limit of a sequence using logarithm properties
MATH 152: Sequences Exercise 7
Finding the limit of a sequence
Increasing, Decreasing, and Non Monotonic Sequences
Explaining increasing, decreasing, and non monotonic for sequences
MATH 152: Sequences Exercise 10
Determining if a sequence is increasing, decreasing, or non monotonic
The Chain Rule (multivariable) Conceptual V1
Explaining the Chain Rule for multivariable functions
The Chain Rule (multivariable) Exercise V3
Using the Chain Rule for multivariable functions to find the rate a volume is changing
Cases for Partial Fraction Decomposition
Explaining all four cases of partial fraction decomposition
Finding the nth Taylor Polynomial Exercise
Finding the nth Taylor polynomial of the natural log function centered at 1
Approximation Using Taylor Polynomials Exercise
Using a Taylor polynomial to approximate the value of a function and estimating the error
The Chain Rule (multivariable) Conceptual V1
Explaining the Chain Rule for multivariable functions
WIR2 20B M150 V7
Solving a rational equation and checking the solutions
Math 140 Exam 3 Review Exercise 5
Finding the domain, intercepts, holes, and vertical asymptotes for a rational function
Math 140 Exam 3 Review Exercise 7
Compute and simplify the difference quotient for a rational function
Simplifying a Rational Expression
Simplifying a rational expression
Subtracting Rational Expressions
Adding and then simplifying two rational expressions
Multiplying Rational Expressions
Multiplying two rational expressions and then simplifying
Dividing Rational Expressions
Dividing rational expressions and then simplifying the answer
Adding Rational Expressions
Adding two rational expressions and then simplifying the answer
Holes and Asymptotes of Rational Functions
Finding any holes, vertical asymptotes, and horizontal asymptotes of a rational function
Holes and Asymptotes of Rational Functions (2)
Finding any holes, vertical asymptotes, and horizontal asymptotes of a rational function
Matching a Rational Function with its Graph
Matching a rational function with its graph
Domain of a Rational Function
Finding the domain of a rational function
Domain of a Rational Function (2)
Finding the domain of a rational function
Function Composition with a Rational Function
Finding the composition of a rational function and a quadratic
Inverse of a Rational Function
Finding the inverse of a rational function and its domain
Inverse of a Rational Function (2)
Finding the inverse of a rational function and its domain
Difference Quotient of a Rational Function
Finding and simplifying the difference quotient for a rational function
Difference Quotient of a Square Root Function
Finding and simplifying the difference quotient for a square root function
Solving an Inequality with a Rational Function
Solving an inequality with a rational expression by factoring and using a sign diagram
Solving an Inequality with a Sum of Rational Functions
Solving an inequality with two rational expressions using a sign diagram
Solving an Inequality with by Finding a Least Common Denominator
Solving an inequality with a rational expression using a sign diagram
A Composition of Tangent and Arccosine
Evaluating a composition of tangent and arccosine
WIR 20B M152 V51
Determining if a series converges or diverges
WIR 20B M152 V63
Using the Alternating Series Test to determine if a series converges or diverges
Creating and Manipulating Symbolic Expressions in Python
Creating symbolic expression in Python and then factoring, expanding, and simplifying the expressions
Finding and Graphing Tangent Lines to a Curve in Python
Using Python to find the equation of the tangent line to a curve and graphing the result
Finding Derivatives in Python
Finding derivatives in Python and solving for the rate of change of a force
Finding Horizontal Tangent Lines in Python
Finding when the tangent line to a function is horizontal in Python
Finding Intervals of Increase/Decrease and Concavity in Python
Using Python to find the intervals where a function is increasing or decreasing
List Comprehension and Patterns in Higher Derivatives in Python
Using Python's list comprehension tool to find several higher order derivatives of a function at once
Parametric Plots and Tangents at a Given t-value in Python
Using Python to find the tangent line to a parametric equation and plot the two graphs
Tangents to Parametrized Curves at a Given Point in Python
Using Python to plot a parametrized curve and its tangent line at a point
Domain of Rational Functions Video 1
Finding the domain of several rational functions
Domain of Rational Functions Video 2
Finding the domain of a rational function
Domain of Rational Functions Video 3
Explaining how to find any holes and vertical asymptotes of a rational function
Domain of Rational Functions Video 4
Finding the domain of a rational function using factoring
Domain of Combinations of Functions Video 1
Finding the domain of a function with a denominator, square root, and logarithm
Properties of Rational Functions and Expressions Video 1
Discussing parent functions of rational functions and their properties
Properties of Rational Functions and Expressions Video 2
Finding horizontal asymptotes for rational functions
Properties of Rational Functions and Expressions Video 4
Finding horizontal asymptotes for rational functions
Properties of Rational Functions and Expressions Video 5
Determining the end behavior and horizontal asymptotes for rational functions
Properties of Rational Functions and Expressions Video 7
How to simplify rational expressions
Properties of Rational Functions and Expressions Video 8
Adding and subtracting rational expressions by finding a common denominator
Properties of Rational Functions and Expressions Video 9
Explaining how to multiply and divide rational expressions
Properties of Rational Functions and Expressions Video 10
Performing algebraic operations with rational expressions and simplifying the answer
Properties of Rational Functions and Expressions Video 3
Explaining how to do polynomial long division
Solving Rational Equations Video 1
Solving equations that contain rational expressions
Solving Rational Equations Video 2
Solving an equation that contains rational expressions
Solving Rational Equations Video 3
Solving an equation that contains rational expressions
Introduction to Riemann Sums
Estimating the area under a function using right-hand, left-hand, and midpoint Riemann sums
Review for the Common Exam: MATH 151 Exam 1 Review Problems 14-25
Review of limits, continuity, and the Intermediate Value Theorem