Unit 2 Lesson #1 Derivatives 1 Interpretations of the Derivative 1. As the slope of a tangent line to a curve. 2. As a rate of change. The (instantaneous) rate of change of y = f (x ) with respect to x when x = a is equal to f '(a). The (instantaneous) rate of change of y = f (x ) with respect to x when x = a is equal to f '(a). y = f(x) (a, f(a)) The derivative of the curve will determine the slope of the tangent line at any given point. The derivative of the curve will determine the slope of the tangent line at any given point. m = f ' (a)

Derivative Notations 2 Emphasizes the idea that differentiation is an operation performed on f. Emphasizes the idea that differentiation is an operation performed on f. “ dy dx ” or “the derivative of y with respect to x” “ dy dx ” or “the derivative of y with respect to x” Names both variables and uses d for derivative. Names both variables and uses d for derivative. “ d dx of f at x ” or “the derivative of f at x ” “ d dx of f at x ” or “the derivative of f at x ” Brief, names the function and the independent variable Brief, names the function and the independent variable The derivative of f with respect to x The derivative of f with respect to x Nice and brief, but does not name the independent variable. f '(x) “ y prime” y '

Definition of Derivative 3 A function is differentiable at x if its derivative exists at x. The process of finding derivatives is called differentiation. The derivative of f at x is given by

Differentiation Rules The Constant Rule The Constant Rule The Power Rule The Power Rule Constant Multiple Rule Constant Multiple Rule Sum and Difference Rules Sum and Difference Rules 4 It would be time-consuming and tedious if we always had to compute derivatives directly from the definition of a derivative. Fortunately, there are several rules that greatly simplify the task of differentiation.

The Constant Rule 5 If the derivative of a function is its slope, then for a constant function, the derivative must be zero. EXAMPLE 1 : The derivative of a constant is zero. 3 Slope = 0

6 First Principals EXAMPLE 2 Find the derivative of y = x from first principles. EXAMPLE 2 Find the derivative of y = x from first principles.

7 EXAMPLE 3 Find the derivative of y = x 2 from first principles. EXAMPLE 3 Find the derivative of y = x 2 from first principles. h First Principals

8 EXAMPLE 3: THE POWER RULE We saw that if, This is part of a pattern. power rule

EXAMPLE 3: Find the derivative of y = 2x. 9 from first principles using the constant multiple rule

Practise Question 1: Find the derivative of y = 3x 2 from first principles. 10

Constant Multiple Rule: 11 Try These! y = 8x 3 y = – 3 x 5 y' = (3)(8) x 3 – 1 y' = 24 x 2

12 The derivative is the = 7 EXAMPLE 4 : Find the derivative of f (x) = 7x – 4 from first principals EXAMPLE 4 : Find the derivative of f (x) = 7x – 4 from first principals

13 Practise Question 6 Find the derivative of y = 2x 2 – 3x – 5 from first principles.

14 Practise Question 6 Find the derivative of y = 2x 2 – 3x – 5 using Sum and Difference Rules

15 The derivative of any curve will determine the slope of the tangent line at any given point. Find the slope of tangent line when x = 1 Find the equation of the tangent line when x = 1