Introductory calculus i : understanding the derivative

Content: 1. The foundation of the derivative --
The derivative of a function at a point --
The derivative as a function --
The Leibniz notation. 2. Using the derivative for powers and linear combinations --
The derivatives of rational powers of x --
The derivatives of linear combinations --
Higher-order derivatives --
The proof of the power rule for arbitrary rational powers. 3. Using the derivatives of sine and cosine --
The derivatives of sine and cosine at 0 --
The derivative functions corresponding to sine and cosine. 4. Using the derivative in velocity and acceleration. 5. Local linear approximations --
The differential --
The traditional notation for the differential --
The accuracy of local linear approximations. 6. Understanding the product and quotient rules --
The quotient rule. 7. Applying the chain rule --
A plausibility argument for the chain rule --
The chain rule in the Leibniz notation --
The chain rule for more than two functions --
The proof of the chain rule. 8. The problems of related rates. 9. The intermediate value theorem --
Newton\'s method. 10. Using implicit differentiation.