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: 9


$(a)$. $$\frac{dy}{dx}=\frac{20x^3}{\cos y}$$ $(b)$. The slope of the tangent line to the curve is $m=-20$

Work Step by Step

$(a)$ Use implicit differentiation to determine $\frac{dy}{dx}$ for $\sin y=5x^4-5 $ Taking the derivative implicitly we get: $$\cos y\frac{dy}{dx}=20x^3$$ solve for $\frac{dy}{dx}$ $$\frac{dy}{dx}=\frac{20x^3}{\cos y}$$ $(b)$. Find the slope of the tangent line to the curve at $(1, \pi)$ We plug in the point $(1,\pi)$ into the derivative from part $(a)$. Hence: $$\frac{dy}{dx}=\frac{20(1)^3}{\cos (\pi)}=\frac{20}{-1}=-20$$
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