Material Taken From: Mathematics for the international student Mathematical Studies SL Mal Coad, Glen Whiffen, John Owen, Robert Haese, Sandra Haese and Mark Bruce Haese and Haese Publications, 2004

Objectives Describe what the first derivative yields. Find the first derivative of functions. Write equations of the line tangent to a curve.

Introductory Videos Brainpop.com o Calculus Khan Academy o Calculus (Derivatives 1) Section 19A-F – Introduction to Calculus

Consider point A (a, f(a)) and point B (x, f(x)) slope of the tangent line at A

Main Ideas: The derivative of a curve at a point is the slope of the tangent line at that point. The derivative function is the function that represents the slopes of all the tangents throughout the entire curve.

Graphing Example 1

Graphing Example 2 What is the function or equation for this graph? What is the derived function of this graph? What would that function look like? What would that function represent?

name of original functionname of derivative function f(x)f(x)f’(x) yy' y Notation and Terminology Differentiation is the process of finding the derivative or the derivative function (the slope function). Notation

FunctionDerivative Function f(x) = x f’(x) = f(x) = x 2 f’(x) = f(x) = x 3 f’(x) = f(x) = x 4 f’(x) = f(x) = x 5 f’(x) = f(x) = x -1 f’(x) = f(x) = x -2 f’(x) = f(x) = x -3 f’(x) = f(x) = f’(x) = f(x) = f’(x) =

Rules (Page 615) f(x)f’(x)in words: a0 The derivative of a constant is zero. xnxn nx n-1 Bring down the power, subtract one from the power. ax n anx n-1 Multiply by the coefficient (same as above). u(x) + v(x)u’(x) + v’(x) The derivative of the sum is the sum of the derivatives.

Find the first and second derivative of f(x) = 3x 4 + 2x 3 – 5x 2 + 7x + 6 Example 1

Find the first and second derivative of f(x) = 5x 3 + 6x 2 – 3x+ 2 Example 2

Find the derivative of Hint: First rewrite the function, then take the derivative. Example 3

Find the slope function of Hint: First rewrite the function, then take the derivative. Example 4

a) Find the gradient function of and then find: b) gradient of the tangent to the function where x = 2. c) equation of the tangent when x = 2. Example 5

a) Find the gradient function of and then find: The gradient function is the first derivative. Now find the gradient when x = 2. Therefore, m = 5 Finally, find the equation of the line. You need a point. We already have the slope. So, the point is (2, 2) and the slope is 5. Therefore, the equation of the tangent at x = 2 is y = 5x – 8. b) gradient of the tangent to the function where x = 2. c) equation of the tangent when x = 2. Example 5 Answer

Find the equation of the tangent to f(x) = x at the point where x = 1 Example 6

Find the equations of any horizontal tangents to y = x 3 – 12x + 2 Example 7

Summary The rate of change at a point is the slope of the tangent line at that point. The slope of the tangent line at that point is known as the derivative. Therefore: – rate of change = slope of tangent line = derivative

Homework 19E, #1a-l, 2a-f 19F #1a-d, 2abc, 3abc Kahn Academy Video – Calculus: Derivatives 3