2. DIFFERENTIATION Differentiation is about rates of change. Differentiation is all about finding rates of change of one quantity compared to another. We need differentiation when the rate of change is not constant.

3. Constant Rate of Change Let's take an example of a car travelling at a constant 60 km/h. The distance-time graph would look like this: We notice that the distance from the starting point increases at a constant rate of 60 km each hour, so after 5 hours we have travelled 300 km. We notice that the slope (gradient) is always 300/5 = 60 for the whole graph. There is a constant rate of change of the distance compared to the time. The slope is positive all the way (the graph goes up as you go left to right along the graph.)

4. Rate of Change that is Not Constant Now let's throw a ball straight up in the air. Because gravity acts on the ball it slows down, then it reverses direction and starts to fall. All the time during this motion the velocity is changing. It goes from positive (when the ball is going up), slows down to zero, then becomes negative (as the ball is coming down).

5. Cont’d Notice this time that the slope of the graph is changing throughout the motion. At the beginning, it has a steep positive slope (indicating the large velocity we give it when we throw it). Then, as it slows, the slope get less and less until it become 0 (when the ball is at the highest point and the velocity is zero). Then the ball starts to fall and the slope becomes negative (corresponding to the negative velocity) and the slope becomes steeper (as the velocity increases).

6. Important Concept – Approximations (limit) of the Slope Notice that if we zoom in close enough to a curve, it begins to look like a straight line. We can find a very good approximation to the slope of the curve at the point t = 1 (it will be the slope of the tangent to the curve, marked in dark red) by observing the points that the curve passes through near t = 1. (A tangent is a line that touches the curve at one point only.)

7. Cont’d Observing the graph, we see that it passes through (0.9, 36.2) and (1.1, 42). So the slope of the tangent at t = 1 is about: The units are m/s, as this is a velocity. We have found the rate of change by looking at the slope.

8. Tangent line Clearly, if we were to zoom in closer, our curve would look even more straight and we could get an even better approximation for the slope of the curve. This idea of "zooming in" on the graph and getting closer and closer to get a better approximation for the slope of the curve (thus giving us the rate of change) was the breakthrough that led to the development of differentiation. The slope of the straight line through is given by

9. Example 1: Find the equation of the tangent line to the curve, at the point. Solution

10. The derivative (or differentiation) of a function at a point using tangent line, is defined as: provided the limit exists and is a small increment. The process of getting the tangent line is called differentiation from the first principles differentiation using definition

11. Example 2 Find the derivate of at using the First Principle Method. Solution

12. Example 3 Find the derivative of the function, using the First Principle. Solution

13. Find the derivatives of the following functions using definitions.

14. Differentiation Formulas or where c is constant function. f(x) = 180 or where n is any number either integer or rational or where c is constant function.

15. Power Rule

16. Exercise

17. Derivatives of Trigonometric Functions (Derivative of the sine function) (Derivative of the cosine function) (Derivative of the tangent function) (Derivative of the secant function) (Derivative of the cotangent function) (Derivative of the cosecant function)

18. Example

19. Example 12 Differentiate of following functions SOLUTION

20. Differentiate of following functions SOLUTION

21. Logarithmic Differentiation where

22. Example 13 Differentiate the following functions with respect to x. Solution where

23. Differentiate the following functions with respect to x. (using the product rule) Solution

24. Differentiate the following functions with respect to x. Solution

25. Differentiations of the exponential function where

26. Example 14 Find for each of the following exponential function Solution

27. Find for each of the following exponential function Solution

28. Find for each of the following exponential function Solution

29. Chain Rule Suppose that we have two functions f(x) and g(x) and they are both differentiable. 1 ) If we define then the derivative of F(x) is, 2 ) If we have y=f(u) and u=g(x) then the derivative of y is,

30. EXAMPLE 9 Find if Solution Let So, then we get Also By using Chain Rule we get,

31. Chain Rule Suppose that we have two functions f(x) and g(x) and they are both differentiable. 1 ) If we define then the derivative of F(x) is, 2 ) If we have y=f(u) and u=g(x) then the derivative of y is,

32. EXAMPLE 9 Find if Solution Let So, then we get Also By using Chain Rule we get,

36. Revisions : Find the derivation

41. Implicit differentiations The equation Is define explicitly as for Is define implicitly as for and when y is differentiable function of x.