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

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

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

Differentiation Rules • The Constant Rule • The Power Rule • Constant Multiple Rule • Sum and Difference Rules 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.

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

EXAMPLE 2 Find the derivative of y = xfrom first principles. Power Rule First Principals

EXAMPLE 3 Find the derivative of y = x2 from first principles. First Principals h

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

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

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

Constant Multiple Rule: Try These! y = 8x 3 y’ = (3)(8) x 3 – 1 y’ = 24 x 2 = – 15 x 4 y = – 3 x5 = (5) (– 3) x 5 – 1 = 10 x g

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

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

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

The derivative of any curve will determine the slope of the tangentline at any given point. EXAMPLE 5: Find the derivative of Find the slope of tangent line when x = 1 Find the equation of the tangent line when x = 1