`e^y sin(x) = x + xy` Find `(dy/dx)` by implicit differentiation.

Textbook Question

Chapter 3, 3.5 - Problem 14 - Calculus: Early Transcendentals (7th Edition, James Stewart).
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hkj1385 | (Level 1) Assistant Educator

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Note:- 1) If  y = sinx ; then dy/dx = cosx 

2) If y = e^x ; then dy/dx = e^x

3) If y = u*v ; where both u & v are functions of 'x' , then

dy/dx = u*(dv/dx) + v*(du/dx)

4) If y = x^n ; where 'k' = constant ; then dy/dx = n*x^(n-1)

Now, the given function is :-

(e^y)*sinx = x + xy

Differentiating both sides w.r.t 'x' we get

(e^y)*cosx + {(e^y)*sinx}*(dy/dx) = 1 + y + x*(dy/dx)

or, dy/dx = [e^y)*cosx - 1 - y]/[x - {(e^y)*sinx}]

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