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Section 3.3a

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The Do Now Find the derivative of Does this make sense graphically???

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The Do Now Find the derivative of Lets see some patterns so we can generalize some easier rules for finding these derivatives…

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Suppose is a function with a constant value c. Then Rule 1: Derivative of a Constant Function If is the function with the constant value c, then

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If, then the difference quotient for is: Recall the algebraic identity: Here, we will let and : What happens when we let h 0?

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Rule 2: Power Rule for Integer Powers of x If is an integer, then The Power Rule says: To differentiate x, multiply by n and subtract 1 from the exponent. n

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Suppose we have a function comprised of a differentiable function of x multiplied by a constant: Then Rule 3: The Constant Multiple Rule If is a differentiable function of x and c is a constant, then The constant goes along for the ride

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Returning to the Do Now Find the derivative of

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What if we have a function comprised of a sum of other differentiable functions? The derivative:

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Rule 4: The Sum and Difference Rule 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, The derivative of a sum (or difference) is the sum (or difference) of the derivatives

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Guided Practice Findif Take each term separately:

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Guided Practice Find if With no negative exponents:

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Second and Higher Order Derivatives The derivative is called the first derivative of y with respect to x. The first derivative may itself be a differentiable function of x. If so, its derivative is called the second derivative of y with respect to x. If (y double-prime) is differentiable, its derivative is called the third derivative of y with respect to x. The pattern continues…

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Second and Higher Order Derivatives However, the multiple-prime notation gets too cumbersome after three primes. We use to denote the n-th derivative of y with respect to x. (We also use.) Note: Do not confuse with the n-th power of y, which is

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Guided Practice Find the first four derivatives of. First derivative: Second derivative: Third derivative: Fourth derivative:

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Guided Practice Does the curve have any horizontal tangents? If so, where? If there are any horizontal tangents, they occur where the slope dy/dx is zero… Solve the equation dy/dx = 0 for x: How can we support our answer graphically?

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Finding f '(x) from the Definition of the Derivative (page 643) The four steps used to find the derivative f ' (x) for a function y = f (x) Are summarized.

Finding f '(x) from the Definition of the Derivative (page 643) The four steps used to find the derivative f ' (x) for a function y = f (x) Are summarized.

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