Explanation Given: Function f ( x ) = 4 x 2 + 1 Formula Used: Definition of derivative is stated as follows: f ′ ( x ) = lim h → 0 f ( x + h ) − f ( x ) h Calculation: Function is given as f ( x ) = 4 x 2 + 1 According to the definition of derivative, we have f ′ ( x ) = lim h → 0 [ 4 ( x + h ) 2 + 1 ] − [ 4 x 2 + 1 ] h f ′ ( x ) = lim h → 0 [ 4 ( x 2 + 2 x h + h 2 ) + 1 ] − [ 4 x 2 + 1 ] h f ′ ( x ) = lim h → 0 [ 4 x 2 + 8 x h + 4 h 2 + 1 ] −

6th Edition

ISBN: 9781119275565

Author: Hughes-Hallett

Publisher: WILEY

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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 3, Problem 5FOT

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Show that f′(x)=8x for the function f(x)=4x2+1

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Show that f ′ ( x ) = 8 x for the function f ( x ) = 4 x 2 + 1

Solution Summary: The author explains that the function f(x) is 8x. The definition of derivative is stated as follows.

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Show that f′(x)=8x for the function f(x)=4x2+1

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Chapter 3 Solutions