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: 28



Work Step by Step

Given $$y^{2} x^{3}+y^{3} x^{4}-10 x+y=5, \ \ \ p( 2,1)$$ Differentiate with respect to $x$ \begin{align*} \frac{d}{d x}\left(y^{2} x^{3}+y^{3} x^{4}-10 x+y\right)&=\frac{d}{d x}(5)\\ 2 y x^{3} y^{\prime}+3 y^{2} x^{4} y^{\prime}+y^{\prime}-10+3 x^{2} y^{2}+4 x^{3} y^{3}&=0\\ \left(2 y x^{3}+3 y^{2} x^{4}+1\right) y^{\prime}&=10-3 x^{2} y^{2}-4 x^{3} y^{3} \end{align*} Then \begin{aligned} y^{\prime} &=\frac{\left(10-3 x^{2} y^{2}-4 x^{3} y^{3}\right)}{\left(2 y x^{3}+3 y^{2} x^{4}+1\right)} \\ \left.y^{\prime}\right|_{(2,1)} &=\frac{(10-12-32)}{(16+48+1)} \\ &=-\frac{34}{65} \end{aligned}
Update this answer!

You can help us out by revising, improving and updating this answer.

Update this answer

After you claim an answer you’ll have 24 hours to send in a draft. An editor will review the submission and either publish your submission or provide feedback.