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**3.3 Rules for Differentiation**

AKA “Shortcuts”

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**Review from 3.2 4 places derivatives do not exist: Corner Cusp**

Vertical tangent (where derivative is undefined) Discontinuity (jump, hole, vertical asymptote, infinite oscillation) In other words, a function is differentiable everywhere in its domain if its graph is smooth and continuous.

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**3.2 Intermediate Value Theorem for Derivatives**

If a and b are any two points in an interval on which f is differentiable, then f’ takes on every value between f’(a) and f’(b).

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**Derivatives of Constants**

Find the derivative of f(x) = 5. Derivative of a Constant: If f is the function with the constant value c, then, (the derivative of any constant is 0)

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**Power Rule If n is any real number and x ≠ 0, then**

What is the derivative of f(x) = x3? From class the other day, we know f’(x) = 3x2. If n is any real number and x ≠ 0, then In other words, to take the derivative of a term with a power, move the power down front and subtract 1 from the exponent.

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**Power Rule Example: Example: What is the derivative of**

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Power Rule Example: What is the derivative of Now, use power rule

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**Constant Multiple Rule**

Find the derivative of f(x) = 3x2. Constant Multiple Rule: If u is a differentiable function of x and c is a constant, then In other words, take the derivative of the function and multiply it by the constant.

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**Sum/Difference Rule Find the derivative of f(x) = 3x2 + x**

If u and v are differentiable functions of x, then their sum and difference are differentiable at every point where u and v are differentiable. At such points, In other words, if functions are separated by + or –, take the derivative of each term one at a time.

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Example Find where horizontal tangent occurs for the function f(x) = 3x3 + 4x2 – 1. A horizontal tangent occurs when the slope (derivative) equals 0.

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**Example At what points do the horizontal tangents of**

f(x)=0.2x4 – 0.7x3 – 2x2 + 5x + 4 occur? Horizontal tangents occur when f’(x) = 0 To find when this polynomial = 0, graph it and find the roots. Substituting these x-values back into the original equation gives us the points (-1.862, ), (0.948, 6.508), (3.539, )

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**Product Rule If u and v are two differentiable functions, then**

Also written as: In other words, the derivative of a product of two functions is “1st times the derivative of the 2nd plus the 2nd times the derivative of the 1st.”

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Product Rule Example: Find the derivative of

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Quotient Rule If u and v are two differentiable functions and v ≠ 0, then Also written as: In other words, the derivative of a quotient of two functions is “low d-high minus high d-low all over low low.”

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Quotient Rule Example: Find the derivative of

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**Higher-Order Derivatives**

f’ is called the first derivative of f f(n) is called the nth derivative of f f'' is called the second derivative of f f''' is called the third derivative of f

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**Higher-Order Derivatives**

Example Find the first four derivatives of

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**Friday Classwork: Section 3.3**

(#1-9 odd, odd, 25, 27, 29, odd, 46)

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Product and Quotient Rules. Product Rule Many people are tempted to say that the derivative of the product is equal to the product of the derivatives.

Product and Quotient Rules. Product Rule Many people are tempted to say that the derivative of the product is equal to the product of the derivatives.

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