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Calculus: An Applied Approach (Min...

10th Edition
Ron Larson
ISBN: 9781305860919

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Calculus: An Applied Approach (Min...

10th Edition
Ron Larson
ISBN: 9781305860919
Textbook Problem
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Finding Derivatives Implicitly In Exercises 53-56, find dy/dx implicitly.

4 x y + ln ( x 2 y ) = 7

To determine

To calculate: The value of dydx for the equation 4xy+ln(x2y)=7 using the implicit differentiation.

Explanation

Given information:

The provided equation is 4xy+ln(x2y)=7.

Formula used:

Product rule of derivative of differentiable functions, f(x) and g(x) is:

ddx[f(x)g(x)]=f(x)g(x)+f(x)g(x).

Let u be a differentiable function of x then ddx[lnx]=1x,x>0 and ddx(lnu)=1ududx,u>0.

The derivative of function f(x)=un using the chain rule is:

f(x)=ddx(un)=nun1dudx

Where, u is the function of x.

Calculation:

Consider the equation,

4xy+ln(x2y)=7

Take ddx of both sides of the above equation,

ddx(4xy+ln(x2y))=ddx(7)ddx(4xy)+ddx(lnx2y)=ddx(7)

Apply the product rule of derivative to the function 4xy and x2y,

(ddx(4xy)+dydx(lnx2)+dydx(lny))=dydx(7)4(xdy

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