# The derivative of the function 4 cos x sin y = 1 by implicit differentiation. ### Single Variable Calculus: Concepts...

4th Edition
James Stewart
Publisher: Cengage Learning
ISBN: 9781337687805 ### Single Variable Calculus: Concepts...

4th Edition
James Stewart
Publisher: Cengage Learning
ISBN: 9781337687805

#### Solutions

Chapter 3.5, Problem 11E
To determine

## To calculate: The derivative of the function 4cosxsiny=1 by implicit differentiation.

Expert Solution

The derivative of the function is tanxtany .

### Explanation of Solution

Given information:

The function 4cosxsiny=1 .

Formula used:

Thechain rule for differentiation is if f is a function of gthen ddx(f(g(x)))=f'(g(x))g'(x) .

Product rule for differentiation is ddx(fg)=f'(x)g(x)+f(x)g'(x) where f and g are functions of x .

Calculation:

Consider the function 4cosxsiny=1 .

Differentiate both sides with respect to x ,

ddx(4cosxsiny)=ddx(1)

Recall that chain rule for differentiation is if f is a function of gthen ddx(f(g(x)))=f'(g(x))g'(x) .

Also for the terms of the above expression, apply the product rule for differentiation.

Recall that product rule for differentiation is ddx(fg)=f'(x)g(x)+f(x)g'(x) where f and g are functions of x .

Apply it. Also observe that y is a function of x,

ddx(4cosxsiny)=ddx(1)4cosxcosyy'4sinxsiny=0cosxcosyy'sinxsiny=0cosxcosyy'=sinxsiny

Isolate the value of y' on right hand side and simplify,

cosxcosyy'=sinxsinyy'=sinxsinycosxcosyy'=tanxtany

Thus, the derivative of the function is tanxtany .

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