Power Rule for Derivatives: MATH 142
Using the power rule to find derivatives
Problem: Find the domain of the function \(f(x)=\dfrac{x^3-9x}{\sqrt{3x+3}}.\)
Using the power rule to find derivatives
Reviewing the chain rule and the derivatives and limits of trigonometric functions
Related rates problems, differentials, linear and quadratic approximations
Approximation and Newton's Method, and limits and derivatives of exponential functions
Approximation and Newton's Method, and limits and derivatives of exponential functions
Riemann sums, including approximating areas under curves.
Using Reimann sums and the Fundamental Theorem of Calculus
Using Reimann sums and the Fundamental Theorem of Calculus
Examples reviewing integration techniques from Calculus I
Review of work and average value
Examples finding a volume using disks, washers, and by slicing
Reviewing the chain rule and the derivatives and limits of trigonometric functions
Related rates problems, differentials, linear and quadratic approximations
Approximation and Newton's Method, and limits and derivatives of exponential functions
Approximation and Newton's Method, and limits and derivatives of exponential functions
Riemann sums, including approximating areas under curves.
The definition and properties of definite integrals
Using Reimann sums and the Fundamental Theorem of Calculus
Using Reimann sums and the Fundamental Theorem of Calculus
Examples reviewing integration techniques from Calculus I
Review of work and average value
Examples finding a volume using disks, washers, and by slicing
Finding a vector equation for the tangent line to a three-dimensional vector function
Explaining domain restrictions for denominators, even roots, and logarithms
Finding the domain of a function with a denominator, square root, and logarithm
Finding the domain of even and odd roots
Solving quadratic equations with the difference of two squares formula
Solving equations with multiple radicals
Explaining the general properties of even and odd root functions
How to rewrite radical terms as power terms
Explaining how to solve radical equations and then solving example problems
Solving equations with even roots and checking the solutions
Solving an equation that contains rational expressions
Solving an equation that contains rational expressions
Finding horizontal asymptotes for rational functions
Explaining how to do polynomial long division
Finding horizontal asymptotes for rational functions
Determining the end behavior and horizontal asymptotes for rational functions
Determining if functions are polynomials and finding their properties
Determining the end behavior, real zeros, domain, and y-intercept for a polynomial
Determining the end behavior, real zeros, domain, and y-intercept for a polynomial
Determining the end behavior, real zeros, domain, and y-intercept for a polynomial
Determining the end behavior, real zeros, domain, and y-intercept for a polynomial
Determining the end behavior, real zeros, domain, and y-intercept for a polynomial
Finding the domain of a piecewise function and evaluating it
Finding the domain of a function that is a fraction containing exponentials and a square root
Identifying the parent function and transformations for a given graph
Finding the difference quotient for a quadratic function
Review of continuous functions, the limit definition of the derivative, and instantaneous rate of change
Review of limits at infinity and continuity of functions
Review of continuous functions, the limit definition of the derivative, and instantaneous rate of change
Finding a tangent line and approximating marginal cost
Review of derivative rules such as product, quotient, and chain rule
Finding marginal revenue and approximating the cost of the next item
Absolute extrema on closed intervals
Using the first and second derivative to sketch the graph of the function
Determining where functions are continuous
Finding average and instantaneous rates of change and secant and tangent lines
Finding average and instantaneous rates of change and secant and tangent lines
Using derivatives to find instantaneous rate of change
Determining a functions concavity and inflection points
Determining concavity and inflection points and sketching the graph
Using the Derivative to Sketch the Graph of a Function
Using Critical Values to Find Absolute Extrema on Closed Intervals
Finding Absolute Extrema on Open Intervals
Examples solving Definite Integrals
Solving examples finding the area between curves
Finding the area between two curves
Using the Intermediate Value Theorem and Bisection Method to approximate the solution of an equation
Using derivatives to find the velocity and acceleration functions from the position function
Calculating the derivative of a function with exponentials using the chain and product rule
Simplifying an expression with radicals
Rationalizing the denominator of a fraction with square roots
Rationalizing the denominator of a fraction with square roots
Factoring a polynomial
Factoring a quadratic
Dividing rational expressions and simplifying
Subtracting rational expressions and simplifying
Performing operations with rational expressions and simplifying
Performing operations with rational expressions and simplifying
Solving a polynomial equation using factoring by grouping
Finding the domains of functions
Finding the domains of expressions
Finding the end behavior of a polynomial
Finding the end behavior of a polynomial
Finding the end behavior of a polynomial
Finding properties of a polynomial including zeros and end behavior
Finding properties of a polynomial including zeros and end behavior
Finding properties of a polynomial including zeros and end behavior
Using properties of a polynomial to find its graph
Calculating polynomial long division
Solving a nonlinear inequality
Calculating the difference quotient for function with a square root
Cartesian equations and parametric equations of curves
Review of the limit definition of a derivative and calculating the derivative
Review of limits and derivatives of inverse trigonometric functions
Review of the limit definition of a derivative and calculating the derivative
Review of limits and derivatives of inverse trigonometric functions
Mean Value Theorem and using derivatives to find the shape of curves
Antiderivatives and physics applications
Evaluating limits of functions
Evaluating Limits of Functions
Review of the limit definition of a derivative and calculating the derivative
Review of the limit definition of a derivative and calculating the derivative
Derivatives of trigonometric functions and using the Chain Rule
Derivatives of trigonometric functions and using the Chain Rule
Derivatives of exponential and logarithmic functions and the exponential model
Derivatives of exponential and logarithmic functions and the exponential model
Using derivatives to find properties of graphs
Mean Value Theorem and properties of a graph
Mean Value Theorem and properties of a graph
Using derivatives to find properties of graphs
Examples of integration by substitution
Finding the derivative of a function using the limit definition
Differentiating functions using the Power Rule
Differentiating a function using the Power Rule
Differentiating a function with an exponential function and using the Power Rule
Using the Product Rule to differentiate a function
Using the Quotient Rule to differentiate a rational function
Using the Product Rule to differentiate a function
Using the Product and Quotient Rules to differentiate a function
Using the Product Rule to find a first and second derivative
Using the Chain Rule to differentiate functions
Using a linear approximation to approximate the value of a function
Finding the antiderivative of a polynomial
Finding the antiderivative of a polynomial
Finding a function from its second derivative using antidifferentiation
Antidifferentiating to find a function from its derivative
Antidifferentiating twice to find a function from its second derivative
Using antidifferentiation to find the height of a cliff from the impact speed of a dropped stone
Finding the antiderivative of a polynomial
Antidifferentiating twice to find a function from its second d
Antidifferentiating twice to find a function from its second d
Evaluating a definite integral containing a polynomial
Using u-substitution to evaluate an indefinite integral
Using u-substitution to evaluate an indefinite integral
Reviewing u-Substitution
Review of finding the radius and interval of convergence for a power series
Review of Taylor and Maclaurin Series and their properties
Review of Taylor Polynomials and Taylor's Inequality
Finding a Taylor Polynomial for a function
Examples of integration by substitution
Using integrals to solve work problems and find the average value
Examples using integration by parts
Integrating a function with a square root using u-substitution
Finding the area of a region enclosed by two curves
Finding the area of a region bounded by curves
Finding the volume of a solid of revolution using the washer method
Finding the volume of a solid of revolution using cylindrical shells
Using the comparison test to determine if an improper integral converges or diverges
Using the comparison test to determine if an improper integral converges or diverges
Converting parametric equations into a Cartesian equation and graphing
Cartesian equations and parametric equations of curves
Using the limit definition to find derivatives of functions and vector functions
Calculating slopes of tangent lines to parametric curves
Review of the limit definition of a derivative and calculating the derivative
Review of limits and derivatives of inverse trigonometric functions
Review of the limit definition of a derivative and calculating the derivative
Review of limits and derivatives of inverse trigonometric functions
Mean Value Theorem and using derivatives to find the shape of curves
Antiderivatives and physics applications
Evaluating limits of functions
Proving a property of scalar multiplication for limits using the epsilon-delta definition and using the Squeeze Theorem for Limits.
Evaluating limits of functions and using the Squeeze Theorem
Proving a piecewise function and a polynomial are continuous
Proving a product of continuous functions is continuous and using the Intermediate Value Theorem
Using the limit definition to find derivatives of functions and vector functions
Using the limit definition to find derivatives of functions and vector functions
Review of the limit definition of a derivative and calculating the derivative
Review of the limit definition of a derivative and calculating the derivative
Derivatives of trigonometric functions and using the Chain Rule
Using the Chain Rule to prove facts about derivatives
Derivatives of trigonometric functions and using the Chain Rule
Using implicit differentiation to solve a problem and to prove the power rule for rational exponents
Derivatives of exponential and logarithmic functions and the exponential model
Using derivatives to find properties of graphs
Using the Mean Value Theorem to prove facts about functions
Mean Value Theorem and properties of a graph
Mean Value Theorem and properties of a graph
Using derivatives to find properties of graphs
Examples of integration by substitution
Reviewing u-Substitution
Review of finding the radius and interval of convergence for a power series
Review of Taylor and Maclaurin Series and their properties
Review of Taylor Polynomials and Taylor's Inequality
Finding a Taylor Polynomial for a function
Examples of integration by substitution
Using integrals to solve work problems and find the average value
Using integrals to solve work problems and find the average value
Examples using integration by parts
Determining the domain of a three-dimensional vector function
Finding the limit of a three-dimensional vector function
Finding the angle of intersection for two three-dimensional vector functions
Finding and sketching the domain of a function of two variables
Finding and sketching the domain of a function of two variables
Creating symbolic expression in Python and then factoring, expanding, and simplifying the expressions
Graphing a piecewise function using Python
Defining variables in Python and using the variables in equations
Finding the domain of a function with a natural logarithm and denominator
Finding the domain of a function with a root and a logarithm in the denominator
Finding the domain of a logarithmic function
Finding the domain of functions with even and odd roots
Finding the domain of several rational functions
Finding the domain of a rational function
Explaining how to find any holes and vertical asymptotes of a rational function
Finding the domain of a rational function using factoring
Explaining the main three forms for linear equations
Discussing properties of linear functions
Discussing the types of linear graphs including horizontal and vertical lines and their properties
Discussing how to write the equation of a line through two given points
Writing a linear equation of profit from a business word problem
Solving a linear equation algebraically and showing how this relates to the graph
Solving a linear equation algebraically
Solving a linear equation algebraically and showing how this relates to the graph
Explaining the format and possible solutions for a system of two linear equations
Solving a system of linear equations with two variables
Solving a system of two linear equations with two variables using the elimination method
Solving a system of two linear equations using a TI-84 calculator
Solving a system of two linear equations using a TI-84 calculator
Solving a system of two linear equations using a TI-84 calculator
Explaining the standard form for a quadratic equation and the possible number of solutions
Solving quadratic equations by factoring
Solving a quadratic equation by factoring
Solving a quadratic equation by factoring
Solving quadratic equations with the quadratic formula and discussing the number of possible solutions
Explaining when a quadratic equation does not have a real solution
Explaining the graph and properties for the parent function of quadratics
Discussing properties of quadratic functions including the vertex, domain, range, and end behavior
Solving equations with odd radicals
Properties of exponents applied to power and exponential expressions
Properties of multiplying and simplifying power and exponential expressions
Properties of exponents applied to power and exponential expressions
Properties of exponential growth and decay functions
Properties of the exponential function with base e
Simplifying power and exponential expressions using properties of exponents
Simplifying power and exponential expressions using properties of exponents
Simplifying power and exponential expressions using properties of exponents
Simplifying an expression with both power and exponential functions using properties of exponents
Explaining how to solve equations with power and exponential terms
Solving equations with exponential terms
Solving equations with base e exponential terms
Explaining the terminology and rules of logarithms
Explaining the basic properties of logarithmic functions
Expanding logarithmic expressions using logarithm rules
Condensing a logarithmic expression into a single logarithm
Explaining how to solve logarithmic equations
Solving logarithmic equations with a single logarithm term
Solving logarithmic equations with multiple logarithm terms
Solving logarithmic equations with multiple logarithm terms on both sides
Explaining basic algebraic operations for fractions
How to simplify rational expressions
Adding and subtracting rational expressions by finding a common denominator
Explaining how to multiply and divide rational expressions
Performing algebraic operations with rational expressions and simplifying the answer
Solving equations that contain rational expressions
Discussing parent functions of rational functions and their properties
Explaining the format and possible solutions for a system of two linear equations