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Single Variable Calculus: Early Tr...

8th Edition
James Stewart
ISBN: 9781305270343

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BuyFindarrow_forward

Single Variable Calculus: Early Tr...

8th Edition
James Stewart
ISBN: 9781305270343
Textbook Problem

Patients undergo dialysis treatment to remove urea from their blood when their kidneys are not functioning properly. Blood is diverted from the patient through a machine that filters out urea. Under certain conditions, the duration of dialysis required, given that the initial urea concentration is c > I, is given by the equation

t = ln ( 3 c + 9 c 2 8 c 2 )

Calculate the derivative of t with respect to c and interpret it.

To determine

To find: The derivative of t with respect to c with the equation t=ln(3c+9c28c2) and write an interpretation statement based on the result.

Explanation

Given:

The given equation is as below.

t=ln(3c+9c28c2) (1)

Calculation:

Use chain rule to differentiate the equation (1).

Assume the variable u as (3c+9c28c2), and it implies that variable t as lnu.

Use the expression below.

dtdc=dtdu×dudcdtdc=1u×dudc

Substitute (3c+9c28c2) for u.

dtdc=13c+9c28c2×ddc(3c+9c28c2)

dtdc=2[3c+9c28c]×[32+12ddc(9c28c)] (1)

Refer equation (1), separate the term 9c28c.

Indicate it as f(c)=9c28c.

Differentiate f(c)=9c28c using the chain rule.

Assume the variable l as 9c28c, and it implies that variable f(c) as l.

Use the expression below.

df(c)dc=df(c)dl×dldcdf(c)dc=12l×dldc

Substitute 9c28c for l

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