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2.2 Derivatives of Polynomial Functions

2.2 Derivatives of Polynomial Functions. 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)

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2.2 Derivatives of Polynomial Functions

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  1. 2.2 Derivatives of Polynomial Functions 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 TANGENT DISCONTINUITY S. Evans

  2. 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

  3. 2.2 Derivatives of Polynomial Functions Proof of Constant Rule: S. Evans

  4. 2.2 Derivatives of Polynomial Functions Power Rule: If then: where x is one term where n is a real # OR S. Evans

  5. 2.2 Derivatives of Polynomial Functions Proof of Power Rule: S. Evans

  6. 2.2 Derivatives of Polynomial Functions Ex. 1: Differentiate with respect to x: a) S. Evans

  7. 2.2 Derivatives of Polynomial Functions b) S. Evans

  8. 2.2 Derivatives of Polynomial Functions c) S. Evans

  9. 2.2 Derivatives of Polynomial Functions Ex. 2: Find the slope of the tangent line to the curve at x=1 S. Evans

  10. 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

  11. 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

  12. 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

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