Calculus Differentiation Teacher Resources
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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 .
After defining L'Hopital's rule, Sal shows an example of using the rule to solve for the limit as x approaches 0 of (sin x) / x. He also describes what it means for a fraction to have indeterminate form.
After applying L'Hopital's Rule once and showing that the limit is still indeterminate, Sal applies the rule a second and third time before arriving at a limit that exists. He then shows that given that the final limit exists, so do the previous ones including the limit of the first original problem. Note: Practice problems on L'Hopital's rule are available and can be practiced now or after watching the other example videos.
Sal shows two different ways of finding the limit of a rational expression where the limit of the polynomial is of an indeterminate form involving infinity. First, he shows how to solve this using L�Hopital�s Rule and then by using the factoring method to solve for the limit algebraically. Note: Practice problems on L�Hopital�s rule are available and can be practiced now or after watching the other example videos.
In this video, Sal starts with an example where the limit is not indeterminate but rather undefined and therefore cannot be solved using L'Hoptial's rule. He shows how you can rewrite the problem algebraically so that the rule can be applied. Note: Practice problems on L'Hopital's rule are available.
Using the derivative of ln x, the chain rule, and the definition of a limit, Sal shows a proof that derivative of ex = ex. Note: The video titled "Proof of Derivatives of Ln(x) and e^x,Ó has a clearer explanation of this proof.
Using the definition of a limit, various properties of logarithms, and a definition of e, Sal shows the proof of derivative of ln x = 1/x. Note: The video titled "Proof of Derivatives of Ln(x) and e^x,Ó has a clearer explanation of this proof.
Using the definition of a limit, Sal proves the derivative of �x or x1/2 is equal to _ x-1/2.
Using the binomial theorem and definition of a limit, Sal shows a proof that the derivative of xn equals nxn-1.
In a problem where you are not given the original function, but rather only three known points on the graph and a few additional pieces of information about when the first and second derivatives are positive or negative, Sal shows how you can draw an approximate graph of the original function. He also does a quick explanation which proves that a quadratic function has no inflection point.
Using what one learned about finding the minimum and maximum of functions, the optimization problem to find two numbers whose product is -16 and whose sum of squares is a minimum is solved. Sal starts by writing the equations and defining them in terms of only one variable, then finds the minimum point by using the first derivative, and finally proves he has a minimum point by finding the second derivative.
Using a specific example, Sal shows how to find the equation of a tangent line to a given function at a specific point. Specifically, he solves the problem of finding the tangent line to the function f(x) = xex at x = 1. This problem provides a review of the product rule, slope-intercept form of a line, and steps for finding the equation of a line. It also, provides a nice visual understanding of the problem by graphing both the original equation and the found tangent line.
Sal does another optimization example, this time, minimizing the total cost of an open rectangular box. In this problem, the volume can be defined in terms of a single variable and given a cost model, he builds a cost equation. He finds the first derivative and the critical point. Then, he finds the minimum size of the box and leaves the actual cost the materials to the listener to finish.
Before solving any problems, Sal gives an overview of what happens to the slope, first derivative, and second derivative at local maxima and minima to give the viewer a more intuitive feel for these types of problems.
Sal solves an example for finding the rate of change of the height of water in a cone at a specific point when it is being filled at a given rate. In this video, Sal reviews the volume of a cone and the chain rule and then, uses these to find the change in height with respect to time.
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.
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 .
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.
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 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.