Calculus: Early Transcendentals (2nd Edition)

Published by Pearson
ISBN 10: 0321947347
ISBN 13: 978-0-32194-734-5

Chapter 3 - Derivatives - 3.8 Implicit Differentiation - 3.8 Exercises - Page 200: 11


$(a)$ $$\frac{dy}{dx}=-\csc y$$ $(b)$ The slope of the tangent line to the curve is $m=-1$

Work Step by Step

$(a)$ Use implicit differentiation to determine $\frac{dy}{dx}$ for $\cos y=x$ Taking the derivative implicitly we get: $$-\sin y\frac{dy}{dx}=1$$ solve for $\frac{dy}{dx}$ $$\frac{dy}{dx}=\frac{1}{-\sin y}=-\csc y$$ $(b)$ Find the slope of the tangent line to the curve at $(0,\pi/2 )$ We plug in the point $(0,\pi/2 )$ into the derivative from part $(a)$. Hence: $$\frac{dy}{dx}=-\csc (\pi/2)=-1$$
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