Derivatives Teacher Resources
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Calculus: Derivatives 2.5 (new HD version)
By defining the formal definition of a derivative, fï¿½(x), Sal is able to find the general form of the derivate function for the example f(x) = x2. He continues to stress the importance of an intuitive understanding of derivative functions.
Graph And Their Derived Function
Students analyze graphs and determine their general shape. In this calculus instructional activity, students solve functions by taking the derivative, sketch tangent lines and estimate the slope of the line using the derivative. They graph and analyze their answers.
Calculus: Derivatives 3
This video covers the differential notation dy/dx and generalizes the rule for finding the derivative of any polynomial. It also extends the notion of the derivatives covered in the Khan Academy videos, ï¿½Calculus Derivatives 2ï¿½ and ï¿½Calculus Derivatives 2.5 (HD).ï¿½ Note: Additional practice using the power rule for differentiating polynomials (including some with negative exponents) is available to the listener.
Proofs of Derivatives of Ln(x) and e6x
In this video, Sal takes on the challenge of proving both the derivative of ln x = 1/x and of ex = ex, showing that no circular logic is used in the proof. It contains a clearer version of both proofs shown in the videos titled, _Proof d/dx e^x = e^x” and _Proof: d/dx (ln x) = 1/x.”
Calculus: Derivatives 2
Sal continues where he left off with the last video, ï¿½Derivatives 1,ï¿½ by looking at the equation y = x2 and examining the slope of the secant line at a specific point, and again defining the limit as _x approaches zero to get the slope of the tangent line (derivative to the curve) at a specific point. He then generalizes this technique to find the general formula for the slope at any point. Note: This video has similar content to the Khan Academy videos ï¿½Derivatives 2ï¿½ and ï¿½Derivatives 2.5ï¿½ with the ï¿½(new HD version)ï¿½ label, however, the graphs on the HD versions are clearer.
Calculus: Derivatives 1 (new HD version)
Sal defines the term derivative by taking the listener on a well-organized tour of slope. First, he reviews the concept of slope of a line from algebra, then extends this idea to look at the slope of the curve by first examining a secant line to the curve and then using our knowledge of limits, defining the limit as _x approaches zero to get the slope of the tangent line to the curve. Along with the video, there is an interesting practice module that models the connection between the slopes of the tangent lines and the derivative of a function.
Calculus: Derivatives 1
Sal defines the term derivative by taking the listener on a well-organized tour of slope. First, he reviews the concept of slope of a line from algebra, then extends this idea to look at the slope of the curve by first examining a secant line to the curve and then, using our knowledge of limits, defining the limit as _x approaches zero to get the slope of the tangent line to the curve. Along with the video, there is an interesting practice problem that models the connection between the slopes of the tangent lines and the derivative of a function. Note: This video has similar content as the Khan Academy video of the same name with ï¿½(new HD version),ï¿½ however, the graphs on the HD version are clearer.
Calculus: Derivative of x^(x^x)
Sal starts with an example of finding dy/dx of y = x2 and builds to showing the solution to the more complicated implicit differentiation problem of finding the derivative of y in terms of x of y = x ^ x ^ x .
Calculus: Derivatives 2 (new HD version)
Sal continues where he left off with the last video, ï¿½Derivatives 1,ï¿½ by looking at the equation y = x2 and examining the slope of the secant line at a specific point. He continues with defining the limit as _x approaches zero to get the slope of the tangent line (derivative to the curve) at a specific point. In the next Khan Academy video, ï¿½Derivatives 2.5,ï¿½ he will generalize this technique to find the general formula for the slope at any point.
In the first example, instead of actually using the quotient rule, Sal rewrites the denominator as a negative exponent and uses the product rule. In subsequent examples, Sal shows, but does not prove, the derivative of several interesting functions including ex, ln x, sin x, cos x, and tan x.
Functions and Everyday Situations
Use real world scenarios to facilitate discussion of the relationship between variables and how they are represented graphically and analytically. This can work in part as an introduction to functions, as a complete lesson, or as an extension to a unit on the library of functions.
Functions and Derivatives
Learners derive functions given a limit. In this calculus instructional activity, student define the derivative of f at x=a, knowing the derivative is a point or just a number. This assignment requires students to work independently as much as possible.
This is comprehensive lesson that considers many aspects of quadratic functions. It includes using factoring, completing the square and the use of the quadratic formula for finding the zeros of the function (including imaginary roots). It also reverses the whole process by looking at either different graphs of quadratic functions or zeros that are given and challenges the learner to derive the function. This lesson provides an excellent review for the second year algebra student or a multi-lesson unit for the more novice student.
In this derivative functions worksheet, students solve and complete 17 various types of problems. First, they find the slope of the tangent line at a given point. Then, students find the derivative of the given functions. In addition, they find the value of the derivative of the given function at the indicate point.
Building an Explicit Quadratic Function by Composition
Use an activity to illustrate the different forms of a quadratic function. Here, the task asks learners to use composition of given functions to build an explicit function. The process emphasizes the impact of the order of composition and the effect that each composition has on the graph of the function. The problem assumes that students are familiar with the process of completing the square.
Transforming the graph of a function
This activity guides learners through an exploration of common transformations of the graph of a function. Given the graph of a function f(x), students generate the graphs of f(x + c), f(x) + c, and cf(x) for specific values of c. The focus is more abstract than algebraic because no formula for the function is given. The task can be used either for instruction or assessment.
Learners are given a graph of a parabola on a coordinate system, but intercepts and vertex are not labeled. The task is to analyze eight given quadratic functions and determine which ones might possibly be represented by the graph. The focus of the exercise is on determining key features of a graph, such as intercepts, maximum or minimum, and which way the graph opens, from the equation. The activity is best suited for use in Algebra I after studying the different forms of a quadratic equation, or as a review exercise in Algebra II.
Seeing is Believing: Derivatives
Learn to solve functions by taking the derivatives. In this calculus lesson plan, students compare the graph of a derivative to that of the original function.
Two Investigations of Cubic Functions
Through learning about cubic functions, students graph cubic functions on their calculator. Students determine the local maximum and minimum points and the tangent line from the x-intercept to a point on the cubic function. Students graph various shifts in the cubic function and describe its' max. and min. points.
Exploring Functions with Calculus and Dynamic Software
Calculus can be project-based and inquiry-centered by using dynamic software.