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 .
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.
After applying Lï¿½Hopitals 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 exits, so do the previous ones including the limit of the first original problem. Note: Practice problems on Lï¿½Hopitalï¿½s 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 Lï¿½Hopitalï¿½s Rule and then by using the factoring method to solve for the limit algebraically. Note: Practice problems on Lï¿½Hopitalï¿½s 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 Lï¿½Hoptialï¿½s rule. He shows how you can rewrite the problem algebraically so that the rule can be applied. Note: Practice problems on Lï¿½Hopitalï¿½s rule are available.
Sal defines the product rule and then shows two examples of how it is used. He then shows an example of finding the derivative by using both the chain rule and product rule together.
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 ï¿½Proof of Derivatives of Ln(x) and e^x,ï¿½ has a clearer explanation of this proof.
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.
After defining Lï¿½Hopitalï¿½s 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.
Are your calculus pupils aware that they are standing on the shoulders of giants? This lesson provides a big picture view of the connection between differential and integral calculus and throws in a bit of history, as well. Note: The calculus controversy paper is not included but one can find a number of good resources on the Internet regarding the development of calculus and the role of Newton and Leibnez.
Students read an article to explain the reasoning behind theorems. In this calculus lesson, students understand the underlying principles of theorems and how it helps them make sense of the problems. They know why they do what they do in AP Calculus.
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.
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 ï¿½Proof of Derivatives of Ln(x) and e^x,ï¿½ has a clearer explanation of this proof.
Students investigate calculus using the TI Calculator. In this calculus activity, students deepen their understanding of calculus. This activity includes step by step instruction for the TI.
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, "Calculus Derivatives 2Ó and "Calculus Derivatives 2.5 (HD).Ó Note: Additional practice using the power rule for differentiating polynomials (including some with negative exponents) is available to the listener.
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.
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.
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.
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.