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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 -- |