# The derivative of the function x 4 ( x + y ) = y 2 ( 3 x − y ) 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 7E
To determine

## To calculate: The derivative of the function x4(x+y)=y2(3x−y) by implicit differentiation.

Expert Solution

The derivative of the function is 3y25x44x3yx46xy+3y2 .

### Explanation of Solution

Given information:

The function x4(x+y)=y2(3xy) .

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 x4(x+y)=y2(3xy) .

Differentiate both sides with respect to x ,

ddx(x4(x+y))=ddx(y2(3xy))ddx(x5)+ddx(x4y)=ddx(3xy2)ddx(y3)

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 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(x4(x+y))=ddx(y2(3xy))ddx(x5)+ddx(x4y)=ddx(3xy2)ddx(y3)5x4+x4y'+4x3y=3y2+6xyy'3y2y'

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

5x4+x4y'+4x3y=3y2+6xyy'3y2y'(x46xy+3y2)y'=3y25x44x3yy'=3y25x44x3yx46xy+3y2

Thus, the derivative of the function is 3y25x44x3yx46xy+3y2 .

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