Calculus (3rd Edition)

Published by W. H. Freeman
ISBN 10: 1464125260
ISBN 13: 978-1-46412-526-3

Chapter 3 - Differentiation - 3.8 Implicit Differentiation - Exercises - Page 152: 38



Work Step by Step

Given $$x+\sqrt{x}=y^{2}+y^{4}, \quad(1,1) $$ Differentiate both sides \begin{align*} 1+\frac{1}{2\sqrt x} &=2yy'+4y^3y' \\ (2y+4y^3)y'&=1+\frac{1}{2\sqrt x} \\ y'&= \frac{1}{(2y+4y^3)} +\frac{1}{2(2y+4y^3)\sqrt x} \\ \end{align*} Then $$m = \frac{ 1}{4} $$ Hence the tangent line is given by \begin{align*} \frac{y-y_1}{x-x_1}&=m\\ \frac{y-1}{x-1}&= \frac{1}{4} \\ 4(y-1)&=(x-1)\\ 4y&=x+3\\ y&= \frac{1}{4}x+\frac{3}{4} \end{align*}
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