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MTH 251 – Differential Calculus Chapter 3 – Differentiation Section 3.2 The Derivative as a Function Copyright © 2010 by Ron Wallace, all rights reserved.

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Review – The Derivative at a Point The derivative was defined as the limit of the difference quotient. That is … If x 0 + h = z, then an alternate definition would be … Note that the result of this limit is a number. That is, the derivative at a specific value of x. Remember: x 0 refers to a specific value of x.

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The Derivative as a Function If we do not specify a specific value of x (i.e. use x instead of x 0 ) we get a function called the derivative of f(x). That is, the derivative of f(x) is the function … OR f(x+h) x x+h h f(x) f(x+h) – f(x)

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Derivative Notation All of the following can be used to designate the function that is the derivative of y = f(x) Reminder: The results of these will be a function.

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Derivative at x = a Notation All of the following can be used to designate the derivative of y = f(x) at x = a Reminder: The results of these will be a number.

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Examples … Determine the following derivatives … IMPORTANT! Memorize these 3 results.

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Examples … Determine the following derivatives …

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Sketching the Graph of f ’(x) using the Graph of f(x) Where is the derivative (i.e. slope) zero? Where is the derivative (i.e. slope) positive? Large or small positive? Where is the derivative (i.e. slope) negative? Large or small negative? Where is the derivative (i.e. slope) constant? Function is a line segment. Derivative is a horizontal line segment.

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Sketching the Graph of f ’(x) using the Graph of f(x) Example – Sketch the graph of the derivative of the following function.

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Left & Right Derivatives at a Point If in the definition of the derivative at a point, you use just the left or right hand limit, the derivative at a point can be considered from just one side or the other. Right-Hand Derivative at x 0 Left-Hand Derivative at x 0 If these are equal, then …

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Left & Right Derivatives at a Point Example:

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Where does a derivative NOT exist? Corner left & right derivatives are different

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Where does a derivative NOT exist? Corner Cusp left & right derivatives are approaching & –

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Where does a derivative NOT exist? Corner Cusp Vertical Tangent The derivative limit is or –

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Where does a derivative NOT exist? Corner Cusp Vertical Tangent Discontinuity see the next theorem

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Differentiability & Continuity If f ’(c) exists, then f(x) is continuous at x = c. Proof …

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Differentiability & Continuity If f ’(c) exists, then f(x) is continuous at x = c. Or … the contrapositive implies … If f(x) is NOT continuous at x = c, then f ’(c) does not exist. NOTES If the derivative does not exist, that does not mean the function is not continuous. If the function is continuous, that does not mean that the derivative exists. Example … the Absolute Value function.

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