1. Bell-Ringer

2. Topic 7 – Introduction to Differential Calculus
Chapter 20: Differential Calculus
Chapter 21: Applications of Differential Calculus

3. Differential Calculus
Chapter 20

4. Rates of Change
A rate is a comparison between two quantities of different kinds.
Calculate the rate using the slope formula: m= 𝑦 2 − 𝑦 1 𝑥 2 − 𝑥 1

5. To find the average rate of change between two points, find the gradient of the line segment between them.
Example 2:

6. To find the instantaneous rate of change (rate of change in a particular instance), find the gradient of the tangent to the graph at that point.
Calculate the instantaneous rate of change using the given point and by plugging in (x+h) for x.
Example:

7. The Derivative Function
Since the gradient of the tangent changes as we move along the graph, we can describe a gradient function for the graph. This is called the derived function or the derivative function of the curve.
We represent the derivative function by 𝑑𝑦 𝑑𝑥 given y in terms of x.
We represent the derivative function by f’(x) given the function f(x).

8. Example:

9. Rules of Differentiation
Differentiation is the process of finding a derivative or gradient function.
Rules for differentiating:

10. Example:

11. Equations of Tangents

12. Example:

13. Normals to Curves

14. Example: