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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

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

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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)

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2.2 Derivatives of Polynomial Functions

Proof of Constant Rule:

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2.2 Derivatives of Polynomial Functions

Power Rule:

If then:

where x is one term

where n is a real #

OR

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2 2 derivatives of polynomial functions4
2.2 Derivatives of Polynomial Functions

Proof of Power Rule:

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2 2 derivatives of polynomial functions5
2.2 Derivatives of Polynomial Functions

Ex. 1: Differentiate with respect to x:

a)

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2.2 Derivatives of Polynomial Functions

Ex. 2: Find the slope of the tangent line to the curve at x=1

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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.

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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.

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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.

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