Answered: 32. e = 2y

Algebra & Trigonometry with Analytic Geometry

13th Edition

ISBN:9781133382119

Author:Swokowski

Publisher:Swokowski

Chapter1: Fundamental Concepts Of Algebra

Section1.2: Exponents And Radicals

Problem 44E

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Question

Implicit differentiation Use implicit differentiation to find
dy/dx.

Expert Solution

Step 1

Given function is $e^{x y} = 2 y$ .

Chain rule of differentiation states that:

$\frac{d}{d x} [f (g (x))] = f' (g (x)) \cdot g' (x)$

Product rule of differentiation states that:

$\frac{d}{d x} [f (x) g (x)] = g (x) f' (x) + f (x) g' (x)$

Differentiating the given function with respect to $x$ , using the above rules of differentiation

$\begin{array}{rcl} \frac{d}{d x} [e^{x y}] & = & \frac{d}{d x} (2 y) \\ e^{x y} \cdot \frac{d}{d x} (x y) & = & 2 \frac{d y}{d x} \\ e^{x y} \cdot (x \frac{d y}{d x} + y \cdot 1) & = & 2 \frac{d y}{d x} \\ x e^{x y} \frac{d y}{d x} + y e^{x y} & = & 2 \frac{d y}{d x} \\ 2 \frac{d y}{d x} - x e^{x y} \frac{d y}{d x} & = & y e^{x y} \\ (2 - x e^{x y}) \frac{d y}{d x} & = & y e^{x y} \\ \frac{d y}{d x} & = & \frac{y e^{x y}}{(2 - x e^{x y})} \end{array}$

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