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


Index.