음(sinh-1(x)) = EXAMPLE 4 Prove that + x2 SOLUTION Let y = sinh-(x). Then sinh(y) = x. If we differentiate this equation implicitly with respect to x, we get dy dx = 1. Since cosh?(y) - sinh?(y) = 1 and cosh(y) 2 0, we have cosh(y) = V1 + sinh?(y), so 1. dy dx 1 cosh(y) V1+ sinh?(y)

Calculus: Early Transcendentals
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Author:James Stewart
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Chapter1: Functions And Models
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음(sinh-1(x)) =
EXAMPLE 4 Prove that
+ x2
SOLUTION Let y = sinh-(x). Then sinh(y) = x. If we differentiate this equation implicitly with respect to x, we get
dy
dx
= 1.
Since cosh?(y) - sinh?(y) = 1 and cosh(y) 2 0, we have cosh(y) = V1 + sinh?(y), so
1.
dy
dx
1
cosh(y)
V1+ sinh?(y)
Transcribed Image Text:음(sinh-1(x)) = EXAMPLE 4 Prove that + x2 SOLUTION Let y = sinh-(x). Then sinh(y) = x. If we differentiate this equation implicitly with respect to x, we get dy dx = 1. Since cosh?(y) - sinh?(y) = 1 and cosh(y) 2 0, we have cosh(y) = V1 + sinh?(y), so 1. dy dx 1 cosh(y) V1+ sinh?(y)
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