Differentiate the following implicit function with respect to x :a. xy² -? =- cos (v²)7cos (y²)b. y = -In (Ryx²) + sin(y)e³Rxy + eRYx*c. Find second derivative implicitly for y? + 2RE¬*y = 3R .

Answered: Differentiate the following implicit…

Calculus: Early Transcendentals

8th Edition

ISBN:9781285741550

Author:James Stewart

Publisher:James Stewart

Chapter1: Functions And Models

Section: Chapter Questions

Problem 1RCC: (a) What is a function? What are its domain and range? (b) What is the graph of a function? (c) How...

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Question

Differentiate the following implicit function with respect to x :
a. xy² -? =- cos (v²)
7
cos (y²)
b. y = -In (Ryx²) + sin(y)e³Rxy + eRYx*
c. Find second derivative implicitly for y? + 2RE¬*y = 3R .

Expert Solution

Step 1

To differentiate the given implicit function with respect to $x$ .

a. Given function is $x y^{2} - \frac{7}{y} = \frac{y}{x^{2}} - \cos (y^{2})$

Differentiatitng the given function with respect to $x$ ,

$\begin{array}{rcl} \frac{d}{d x} (x y^{2} - \frac{7}{y}) & = & \frac{d}{d x} (\frac{y}{x^{2}} - \cos (y^{2})) \\ \frac{d}{d x} (x y^{2}) - \frac{d}{d x} (\frac{7}{y}) & = & \frac{d}{d x} (\frac{y}{x^{2}}) - \frac{d}{d x} (\cos (y^{2})) \\ (x \cdot 2 y + y^{2} \cdot 1) - (7 \cdot \frac{- 1}{y^{2}}) \frac{d y}{d x} & = & (\frac{x^{2} \frac{d y}{d x} - y}{x^{4}}) - (- \sin (y^{2}) \cdot 2 y \frac{d y}{d x}) \\ (x^{4} 2 x y + y^{2}) + x^{4} \frac{7}{y^{2}} \frac{d y}{d x} & = & x^{2} \frac{d y}{d x} - y + 2 y \sin (y^{2}) \frac{d y}{d x} \\ [x^{2} - 2 y \sin y - \frac{7 x^{4}}{y^{2}}] \frac{d y}{d x} & = & (2 x^{5} y + y^{2}) \\ \frac{d y}{d x} & = & \frac{(2 x^{5} y + y^{2})}{(x^{2} - 2 y \sin y - \frac{7 x^{4}}{y^{2}})} \end{array}$

Step 2

Given function is $y = - \ln (R y x^{2}) + \sin (y) e^{3 R x y} + \sin (y) e^{3 R x y} + e^{R y} x^{4}$

Differentiating the given function with respect to $x$ $\begin{array}{rcl} \frac{d y}{d x} & = & \frac{d}{d x} [- \ln (R y x^{2}) + \sin (y) e^{3 R x y} + e^{R y} x^{4}] \\ \frac{d y}{d x} & = & \frac{- 1}{R y x^{2}} (R (2 x y + x^{2} \frac{d y}{d x})) + (e^{3 R x y} \cos (y) \frac{d y}{d x} + \sin (y) e^{3 R x y} (3 R (y + x \frac{d y}{d x}))) + (4 x^{3} e^{R y} + x^{4} e^{R y} \frac{d y}{d x}) \\ \frac{d y}{d x} & = & \frac{- 1}{x} - \frac{1}{R y} \frac{d y}{d x} + e^{3 R x y} \cos (y) \frac{d y}{d x} + 3 R y \sin (y) e^{3 R x y} + 3 R x \sin (y) e^{3 R x y} \frac{d y}{d x} \\ \frac{d y}{d x} & = & \frac{- 1}{x} + 3 R y \sin (y) e^{3 R x y} + (\frac{- 1}{R y} e^{3 R x y} \cos (y) + 3 R x \sin (y) e^{3 R x y}) \frac{d y}{d x} \\ (\frac{- 1}{R y} e^{3 R x y} \cos (y) + 3 R x \sin (y) e^{3 R x y} - 1) \frac{d y}{d x} & = & \frac{- 1}{x} + 3 R y \sin (y) e^{3 R x y} \\ \frac{d y}{d x} & = & \frac{\frac{- 1}{x} + 3 R y \sin (y) e^{3 R x y}}{(\frac{- 1}{R y} e^{3 R x y} \cos (y) + 3 R x \sin (y) e^{3 R x y} - 1)} \end{array}$ ,