1. Index FAQ The derivative as the slope of the tangent line (at a point)

2. Index FAQ Video help: MIT!!! http://ocw.mit.edu/courses/mathematics/ 18-01sc-single-variable-calculus-fall- 2010/1.-differentiation/

3. Index FAQ What is a derivative? A function, which gives the: the rate of change of a function in general the slope of the line tangent to the curve in general

4. Index FAQ What is a differential quotient? Just a number! the rate of change of a function at a given point the slope of the line tangent to the curve at a certain point The substitutional value of the derivative

5. Index FAQ The tangent line single point of intersection

6. Index FAQ slope of a secant line a x f(x) f(a) f(a) - f(x) a - x

7. Index FAQ slope of a (closer) secant line ax f(x) f(a) f(a) - f(x) a - x x

8. Index FAQ closer and closer… a

9. Index FAQ watch the slope...

10. Index FAQ watch what x does... a x

11. Index FAQ The slope of the secant line gets closer and closer to the slope of the tangent line...

12. Index FAQ As the values of x get closer and closer to a! a x

13. Index FAQ The slope of the secant lines gets closer to the slope of the tangent line......as the values of x get closer to a Translates to….

14. Index FAQ lim ax f(x) - f(a) x - a Equation for the slope Which gives us the the exact slope of the line tangent to the curve at a! as x goes to a

15. Index FAQ similarly... a a+h f(a+h) f(a) f(x+h) - f(x) (x+h) - x = f(x+h) - f(x) h (For this particular curve, h is a negative value) h