Derivatives Teacher Resources
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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.
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.
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.
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.
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.”
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.
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.
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.
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 .
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.
Students follow detailed directions to illustrate Rolle's Theorem on their graphing calculator. In this application of derivatives worksheet, students input a given function into their graphing calculator and graph it. They then calculate and graph the derivative of the original function.
Young scholars derive functions given a limit. In this calculus lesson, 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.
In this derivative functions activity, 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.
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 instructional activity, or as an extension to a unit on the library of functions.
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.
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.
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.
Calculus can be project-based and inquiry-centered by using dynamic software.
Students explore how the structure of different plant parts relates to their function. They begin by examining the fastening properties of Velcro?? and comparing them to a method of seed dispersal called hitchhiking.
Familiarize your class with the benzene derivative compounds. Some Sal will recommend memorizing, while others are mainly presented as practice for naming.