# The derivative of the function x 2 + x y − y 2 = 4 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 5E
To determine

## To calculate: The derivative of the function x2+xy−y2=4 by implicit differentiation.

Expert Solution

The derivative of the function is 2x+y2yx .

### Explanation of Solution

Given information:

The function x2+xyy2=4 .

Formula used:

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

Power rule for differentiation is ddxxn=nxn1 .

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 x2+xyy2=4 .

Differentiate both sides with respect to x ,

ddx(x2+xyy2)=ddx(4)ddx(x2)+ddx(xy)ddx(y2)=0

Recall that power rule for differentiation is ddxxn=nxn1 and chain rule for differentiation is if f is a function of gthen ddx(f(g(x)))=f'(g(x))g'(x) .

Also for the second term 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(x2+xyy2)=ddx(4)ddx(x2)+ddx(xy)ddx(y2)=02x+xy'+y2yy'=0xy'2yy'=2xy

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

xy'2yy'=2xy(x2y)y'=2xyy'=2xyx2yy'=2x+y2yx

Thus, the derivative of the function is 2x+y2yx .

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