1. Introduction to Calculus 1: RATE OF CHANGE and FIRST PRINCIPLES
Independent Practice:
Oxford Text:
Exercises p201, 7C; p202, 7D & p302, 7E
Pearson Chapter 11 p

2. Rate of Change (Slope) Average Rate of Change:
Recall: Slope (gradient) of a line can be calculated by:
Average Rate of Change:
Slope of the line (secant) that connects two points on a curve
Average rate of change from A to B = gradient from A to B
Instantaneous Rate of Change:
Slope of the line (tangent) that touches the curve at a point
Instantaneous rate of change at x =1 = gradient of tangent at x =1
A tangent line is difficult to accurately draw. We will explore ways to accurately calculate the slope of a tangent

3. Finding the slope of a secant line
The gradient of the secant line AB is written as
Using the “formula” for the gradient of a secant line, write an expression for the gradient of a secant line for and simplify your answer.

4. Finding the slope of a tangent line
We start by finding the secant slope between two points, one with an x-coordinate “x”, and the other, some distance “h” away, to give a second x-coordinate “x+h”.
We substitute these x – values into the slope formula as follows:
The function defined by is known as the derivative of f.
The notation for the derivative is or so the derivate is defined by
𝑑𝑦 𝑑𝑥 = lim ℎ→0 𝑓 𝑥+ℎ −𝑓(𝑥 ℎ
𝑓′(𝑥)= lim ℎ→0 𝑓 𝑥+ℎ −𝑓(𝑥 ℎ

5. Finding the slope of a tangent using the derivative
1. Find the derivative of and hence find the gradient of the tangent at x = 3.

6. 2. Find the derivatives of the following functions using first principles (in terms of x) (Substitute into the derivative formula & simplify, then find the limit as h ➝ 0)

7. 3. Find the derivatives of the following functions using first principles (in terms of x) (Substitute into the derivative formula & simplify, then find the limit as h ➝ 0)

8. Looking for a general rule
𝑓 𝑥 = 1 𝑥 = 𝑥 −1
𝑓 𝑥 = 1 𝑥 2 = 𝑥 −2
𝑓 𝑥 = 𝑥 = 𝑥 1 2
From the derivatives above can you define a general rule for functions of this type?
If then 𝑓” 𝑥 =
𝑓 𝑥 = 𝑥 𝑛
4. Using the general rule, find the derivatives of the following functions & express your answer with only positive exponents.