Explanation Given: Function is at point x=c. Concept used: First derivative represents slope and tangent it also determines whether the function is increasing or decreasing.it is given by ⇒ Δ y Δ x = f ( x + Δ x ) − y Δ x . Calculation: Lets y = f ( x ) By adding Δ y to left hand side Δ x to right hand side ⇒ y + Δ y = f ( x + Δ x ) ⇒ Δ y = f ( x + Δ x ) − y ⇒ Δ y Δ x = f ( x + Δ x ) − y Δ x So, through here we get first derivative we can re-write as f ′ ( x ) = lim h → 0 f ( x + h ) − f ( x ) h or equivalent to f ′ ( x ) = lim z → x f ( z ) − f ( x ) z − x The domain of f ′ is the set of all value of x for which the defining the limit of f ′ exists. First derivative represents a slope or physics its same a velocity instantaneous change in velocity is given by v ( t ) = s ′ ( t ) For example we can calculate the derivative of f ( x ) = x 2 for all value of x with either a limit of h → 0 or z → x f ′

5th Edition

ISBN: 9781429241861

Author: Laura Taalman, Peter Kohn

Publisher: W. H. Freeman

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Rate of Change

The relation between two quantities which displays how much greater one quantity is than another is called ratio.

Slope

The change in the vertical distances is known as the rise and the change in the horizontal distances is known as the run. So, the rise divided by run is nothing but a slope value. It is calculated with simple algebraic equations as:

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Chapter 2, Problem 5D

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Todefine:The formal definition of the derivative of a function f at point x=c.

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Chapter 2 Solutions

Calculus

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Todefine: The formal definition of the derivative of a function f at point x=c.

Solution Summary: The author explains the formal definition of the derivative of a function f at point x=c. First derivative represents slope and tangent and determines whether the function is increasing or decreasing.

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Todefine:The formal definition of the derivative of a function f at point x=c.

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Chapter 2 Solutions