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Goldstein/Schneider/Lay/Asmar, CALCULUS AND ITS APPLICATIONS, 11e – Slide 1 of 115 § 1.3 The Derivative

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Goldstein/Schneider/Lay/Asmar, CALCULUS AND ITS APPLICATIONS, 11e – Slide 2 of 115 The Derivative Differentiation Slope and the Derivative Equation of the Tangent Line to the Graph of y = f (x) at (a, f (a)) Leibniz Notation for Derivatives Calculating Derivatives Via the Difference Quotient Differentiable Limit Definition of the Derivative Limit Calculation of the Derivative Section Outline

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Goldstein/Schneider/Lay/Asmar, CALCULUS AND ITS APPLICATIONS, 11e – Slide 3 of 115 The Derivative DefinitionExample Derivative: The slope formula for a function y = f (x), denoted: Given the function f (x) = x 3, the derivative is

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Goldstein/Schneider/Lay/Asmar, CALCULUS AND ITS APPLICATIONS, 11e – Slide 4 of 115 Differentiation DefinitionExample Differentiation: The process of computing a derivative. No example will be given at this time since we do not yet know how to compute derivatives. But don’t worry, you’ll soon be able to do basic differentiation in your sleep.

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Goldstein/Schneider/Lay/Asmar, CALCULUS AND ITS APPLICATIONS, 11e – Slide 5 of 115 Differentiation Examples These examples can be summarized by the following rule.

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Goldstein/Schneider/Lay/Asmar, CALCULUS AND ITS APPLICATIONS, 11e – Slide 6 of 115 Differentiation ExamplesEXAMPLE SOLUTION Find the derivative of This is the given equation. Rewrite the denominator as an exponent. Rewrite with a negative exponent. What we’ve done so far has been done for the sole purpose of rewriting the function in the form of f (x) = x r. Use the Power Rule where r = -1/7 and then simplify.

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Goldstein/Schneider/Lay/Asmar, CALCULUS AND ITS APPLICATIONS, 11e – Slide 7 of 115 Differentiation ExamplesEXAMPLE SOLUTION Find the slope of the curve y = x 5 at x = -2. This is the given function. We must first find the derivative of the given function. Use the Power Rule. Since the derivative function yields information about the slope of the original function, we can now use to determine the slope of the original function at x = -2. Replace x with -2. Evaluate. Therefore, the slope of the original function at x = -2 is 80 (or 80/1).

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Goldstein/Schneider/Lay/Asmar, CALCULUS AND ITS APPLICATIONS, 11e – Slide 8 of 115 Equation of the Tangent Line to the Graph of y = f ( x ) at the point ( a, f ( a ))

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Goldstein/Schneider/Lay/Asmar, CALCULUS AND ITS APPLICATIONS, 11e – Slide 9 of 115 Equation of the Tangent LineEXAMPLE SOLUTION Find the equation of the tangent line to the graph of f (x) = 3x at x = 4. This is the given function. We must first find the derivative of the given function. Differentiate. Notice that in this case the derivative function is a constant function, 3. Therefore, at x = 4, or any other value, the value of the derivative will be 3. So now we use the Equation of the Tangent Line that we just saw. This is the Equation of the Tangent Line. f (4) = 12 and Simplify.

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Goldstein/Schneider/Lay/Asmar, CALCULUS AND ITS APPLICATIONS, 11e – Slide 10 of 115 Leibniz Notation for Derivatives Ultimately, this notation is a better and more effective notation for working with derivatives.

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Goldstein/Schneider/Lay/Asmar, CALCULUS AND ITS APPLICATIONS, 11e – Slide 11 of 115 Calculating Derivatives Via the Difference Quotient The Difference Quotient is

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Goldstein/Schneider/Lay/Asmar, CALCULUS AND ITS APPLICATIONS, 11e – Slide 12 of 115 Calculating Derivatives Via the Difference QuotientEXAMPLE SOLUTION Apply the three-step method to compute the derivative of the following function: This is the difference quotient. STEP 1: We calculate the difference quotient and simplify as much as possible. Evaluate f (x + h) and f (x). Simplify.

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Goldstein/Schneider/Lay/Asmar, CALCULUS AND ITS APPLICATIONS, 11e – Slide 13 of 115 Calculating Derivatives Via the Difference Quotient STEP 2: As h approaches zero, the expression -2x – h approaches -2x, and hence the difference quotient approaches -2x. Factor. CONTINUED Cancel and simplify. STEP 3: Since the difference quotient approaches the derivative of f (x) = -x as h approaches zero, we conclude that

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Goldstein/Schneider/Lay/Asmar, CALCULUS AND ITS APPLICATIONS, 11e – Slide 14 of 115 Differentiable DefinitionExample Differentiable: A function f is differentiable at x if approaches some number as h approaches zero. The function f (x) = |x| is differentiable for all values of x except x = 0 since the graph of the function has no definite slope when x = 0 (f is nondifferentiable at x = 0) but does have a definite slope (1 or -1) for every other value of x.

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Goldstein/Schneider/Lay/Asmar, CALCULUS AND ITS APPLICATIONS, 11e – Slide 15 of 115 Limit Definition of the Derivative

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Goldstein/Schneider/Lay/Asmar, CALCULUS AND ITS APPLICATIONS, 11e – Slide 16 of 115 Limit Calculation of the DerivativeEXAMPLE SOLUTION Using limits, apply the three-step method to compute the derivative of the following function: This is the difference quotient. Evaluate f (x + h) and f (x). Simplify. STEP 1:

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Goldstein/Schneider/Lay/Asmar, CALCULUS AND ITS APPLICATIONS, 11e – Slide 17 of 115 Limit Calculation of the Derivative Factor. CONTINUED Cancel and simplify. STEP 2: As h approaches 0, the expression -2x – h approaches -2x STEP 3: Since the difference quotient approaches, we conclude that

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