Explanation Given: The function, f ( x ) = 5 x Concept Used: Slope of the tangent line of a function f ( x ) at any point x = c is given by the first derivative at that point. That is the slope of tangent line at x = c is given by f ′ ( c ) . Slope of the normal line at any point of the curve is equal to the negative reciprocal of the slope of tangent line. That is the slope of tangent line at x = c is given by − 1 f ′ ( c ) . The point-slope form of the line passing through the point ( x 1 , y 1 ) and having slope m is given by y − y 1 = m ( x − x 1 ) . Power rule of derivatives, d d x ( x n ) = n x n − 1 Chain rule of derivatives, d d x ( f ( g ( x ) ) ) = f ' ( g ( x ) ) ⋅ g ' ( x ) Derivative of any constant C is zero, that is, d d x ( C ) = 0 Calculation: Consider the function, f ( x ) = 5 x In order to find the equation of normal line, first we find the slope of the normal line at the specified point. We know that the slope of normal line is negative reciprocal of the slope of tangent line at any point of the curve. And the slope of tangent line is given by first derivative at any point on curve. So, using the power rule and chain rule of derivatives, the first derivative of the function is given by f ' ( x ) = d d x ( 5 x ) 1 2 = 1 2 ( 5 x ) 1 2 − 1 ⋅ d d x ( 5 x ) = 1 2 ( 5 x )

7th Edition

ISBN: 9781524916817

Author: SMITH KARL J, STRAUSS MONTY J, TODA MAGDALENA DANIELE

Publisher: Kendall Hunt Publishing

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Chapter 3.1, Problem 34PS

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The equation of the normal line to the graph of f(x)=5x at x=5 .

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Chapter 3.1, Problem 33PS

Chapter 3.1, Problem 35PS

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Chapter 3 Solutions

Calculus

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The equation of the normal line to the graph of f ( x ) = 5 x at x = 5 .

Solution Summary: The author explains the equation of the normal line to the graph of f(x)=sqrt5x

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The equation of the normal line to the graph of f(x)=5x at x=5 .

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Chapter 3 Solutions