2. Introduction to…. Derivatives

3. Differentiation is one of the most fundamental and powerful operations in all of calculus.

4. It is a concept that was developed over two hundred years ago by two men… Sir Issac Netwon (Lagrange Notation) Gottfried Leibniz (Leibniz Notation)

5. The output of this operation is called the “derivative”. The derivative can be used to calculate the slope of the tangent at any point in the function’s domain.

6. The formula for the derivative is created through the combination of the 2 main concepts we have studies so far 1. The difference quotient 2. Limits

7. Starting with our difference quotient, we no longer want to construct a single secant line starting at “a”. We want to construct a secant line anywhere within the domain…. To do this, we replace “a” with “x”

8. Starting with our difference quotient, we no longer want to construct a single secant line starting at “a”. We want to construct a secant line anywhere within the domain…. To do this, we replace “a” with “x”

9. Next, we no longer want this to represent a secant line. We want a tangent line. We want to know the exact slope at each point “x”. To do this, we must make h infinitely small. A limit will allow us to reduce h in this manner… m tan = h0 f’(x)= Legrange Notation

10. Video of the DerivativeDerivative

11. Find the derivative of f(x) = x 2 (find a function that represents the slopes of all tangents ) f’(x) = lim h 0 lim h 0 lim h 0 lim h 0 = = =

12. lim h 0 = Since h is not zero, we are free to divide 1 1

13. lim h 0 = 2x We can now sub in any value of x to determine the slope of the tangent for every point x in the domain….. …nice Now “take the limit” substitute h = 0 (always try sub First) = Confirm with sketchpad!!!!!

14. Other notation… Leibniz notation Read as “dee y by dee x” It reminds us of the process by which the derivative is obtained D as in delta, as in “the change in y with respect to the change in x”…

15. Use first principles to differentiate f(x) = x 3 f’(x) = lim h 0

16. The height of a javelin tossed in the air is modelled by the function H(t) = -4.9t 2 + 10t + 1, where H is the height, and t is time, in seconds. a)Determine the rate of change of the height of the javelin at time t. b)Determine the rate of change of the javelin after 1,2 and 3 seconds.

17. Note: A function is also non-differentiable at points where the function has an abrupt change, which is represented by a cusp or a corner on the graph.