BuyFind

Single Variable Calculus: Concepts...

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

Single Variable Calculus: Concepts...

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

Solutions

Chapter 3.5, Problem 31E
To determine

To calculate: The second derivative of the function 9x2+y2=9 by implicit differentiation.

Expert Solution

Answer to Problem 31E

The second derivative of the function is y''=81y3 .

Explanation of Solution

Given information:

The function 9x2+y2=9 .

Formula used:

The chain rule for differentiation is if f is a function of g then ddx(f(g(x)))=f'(g(x))g'(x) .

Power rule for differentiation is ddxxn=nxn1 .

Quotient rule for differentiation is ddx(fg)=g(x)f'(x)f(x)g'(x)[g(x)]2 where f and g are functions of x .

Calculation:

Consider the function 9x2+y2=9 .

Differentiate both sides with respect to x ,

  ddx(9x2+y2)=ddx(9)ddx(9x2)+ddx(y2)=0

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

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

  ddx(9x2+y2)=ddx(9)ddx(9x2)+ddx(y2)=018x+2yy'=0

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

  18x+2yy'=02yy'=18xy'=18x2yy'=9xy

Therefore, the value of first derivative of the function is y'=9xy .

Now, again differentiate the above expression with respect to x . Recall that quotient rule for differentiation is ddx(fg)=g(x)f'(x)f(x)g'(x)[g(x)]2 where f and g are functions of x .

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

  ddx(y')=ddx(9xy)y''=(y)ddx(9x)(9x)ddx(y)(y)2y''=y(9)+9x(y')y2y''=9y+9xy'y2

Now, substitute the value y'=9xy in the above expression.

  y''=9y+9x(9xy)y2=9y281x2y3=9(9x2+y2)y3

According to the question 9x2+y2=9 , substitute the value in the above expression,

  y''=9(9x2+y2)y3=9(9)y3=81y3

Thus, the second derivative of the function is y''=81y3 .

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