Differentiate means “find the derivative” A function is said to be differentiable if he derivative exists at a point x=a. NOT Differentiable at x=a means.

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Presentation transcript:

Differentiate means “find the derivative” A function is said to be differentiable if he derivative exists at a point x=a. NOT Differentiable at x=a means that you cannot find the slope of the tangent at x=a. Examples (not differentiable at x=a) CUSP VERTICAL TANGENTDISCONTINUITY S. Evans

2.2 Derivatives of Polynomial Functions Constant rule and Power rule Constant Rule: If where k is a constant then (Prime notation) OR (Leibniz notation) S. Evans

2.2 Derivatives of Polynomial Functions Proof of Constant Rule: S. Evans

2.2 Derivatives of Polynomial Functions Power Rule: If then: where x is one term where n is a real # OR S. Evans

2.2 Derivatives of Polynomial Functions Proof of Power Rule: S. Evans

2.2 Derivatives of Polynomial Functions Ex. 1: Differentiate with respect to x: a) S. Evans

2.2 Derivatives of Polynomial Functions b) S. Evans

2.2 Derivatives of Polynomial Functions c) S. Evans

2.2 Derivatives of Polynomial Functions Ex. 2: Find the slope of the tangent line to the curve at x=1 S. Evans

2.2 Derivatives of Polynomial Functions Ex. 3: Find the co-ordinates of the point(s) on the graph of at which the slope of the tangent is 12. S. Evans

2.2 Derivatives of Polynomial Functions Ex. 4: Tangents are drawn from point (0,-8) to the curve. Find the co-ordinates of the point(s) at which these tangents touch the curve. S. Evans

2.2 Derivatives of Polynomial Functions Vocabulary: Derivative: Also known as instantaneous rate of change with respect to the variable. Displacement, Change in position. Velocity, Rate of change of position with respect to time. Acceleration, Rate of change of velocity with respect to time. S. Evans