Modeling the Dynamics of Life: Calculus and Probability for Life Scientists
Modeling the Dynamics of Life: Calculus and Probability for Life Scientists
3rd Edition
ISBN: 9780840064189
Author: Frederick R. Adler
Publisher: Cengage Learning
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Chapter 2.9, Problem 22E

a.

To determine

To find derivative of the function

  F(x)=(1+2x)3

By expanding the binomial

a.

Expert Solution
Check Mark

Answer to Problem 22E

  F(x)=24x2+24x+6

Explanation of Solution

Given:

  F(x)=(1+2x)3

Concept Used:

  • Addition/subtraction rule of differentiation formula
  • ddx(f(x)±g(x))=ddxf(x)±ddxg(x)

  • Power rule of differentiation formula
  • ddxxn=nxn1

  • (a+b)3=a3+b3+3a2b+3ab2

Calculation:

To find derivative of the function

  F(x)=(1+2x)3

First expand the binomial as

  F(x)=(1+2x)3F(x)=13+(2x)3+3(2x)2(1)+3×12×2xF(x)=1+8x3+12x2+6x

Now, differentiate it, as

  ddx[F(x)]=ddx(1+8x3+12x2+6x)F(x)=ddx1+ddx(8x3)+ddx(12x2)+ddx(6x)F(x)=0+8ddx(x3)+12ddxx2+6ddxxF(x)=8(3x31)+12(2x21)+6(1)F(x)=24x2+24x+6

Hence,

  F(x)=24x2+24x+6

b.

To determine

To find derivative of the function

  F(x)=(1+2x)3

By using chain rule.

b.

Expert Solution
Check Mark

Answer to Problem 22E

  F(x)=24x2+24x+6

Explanation of Solution

Given:

  F(x)=(1+2x)3

Concept Used:

  • Chain rule of differentiation is: ddxf[g(x)]=ddgf(g)dgdx , where g is a differentiable function at x and f is a differentiable function at g(x)
  • Power rule of differentiation formula
  • ddxxn=nxn1

Calculation:

To find derivative of the function

  F(x)=(1+2x)3

Use chain rule of differentiation that is,

  ddxf[g(x)]=ddgf(g)dgdx

Where, f(x)=x3;  g(x)=(1+2x)

Thus, differentiation of given function can be calculated as

  ddx[F(x)]=ddx(1+2x)3F(x)=3(1+2x)31ddx(1+2x)F(x)=3(1+2x)2(ddx1+2ddxx)F(x)=3(1+(2x)2+4x)(0+2(1))F(x)=(3+12x2+12x)2F(x)=24x2+24x+6

Hence,

  F(x)=24x2+24x+6

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Chapter 2 Solutions

Modeling the Dynamics of Life: Calculus and Probability for Life Scientists

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Prob. 24ECh. 2.9 - Prob. 25ECh. 2.9 - Prob. 26ECh. 2.9 - Prob. 27ECh. 2.9 - Prob. 28ECh. 2.9 - Prob. 29ECh. 2.9 - Prob. 30ECh. 2.9 - Prob. 31ECh. 2.9 - Prob. 32ECh. 2.9 - Prob. 33ECh. 2.9 - Prob. 34ECh. 2.9 - Prob. 35ECh. 2.9 - Prob. 36ECh. 2.9 - Prob. 38ECh. 2.9 - Prob. 39ECh. 2.9 - Prob. 40ECh. 2.9 - Prob. 41ECh. 2.9 - Prob. 42ECh. 2.9 - Prob. 43ECh. 2.9 - Prob. 44ECh. 2.9 - The method of implicit differentiation is often...Ch. 2.9 - Prob. 46ECh. 2.9 - Prob. 47ECh. 2.9 - Prob. 48ECh. 2.9 - Prob. 49ECh. 2.9 - Prob. 50ECh. 2.9 - Prob. 51ECh. 2.9 - Prob. 52ECh. 2.9 - Prob. 53ECh. 2.10 - Prob. 1ECh. 2.10 - Prob. 2ECh. 2.10 - Prob. 3ECh. 2.10 - Prob. 4ECh. 2.10 - Prob. 5ECh. 2.10 - Prob. 6ECh. 2.10 - Prob. 7ECh. 2.10 - Prob. 8ECh. 2.10 - Prob. 9ECh. 2.10 - Prob. 10ECh. 2.10 - Prob. 11ECh. 2.10 - Prob. 12ECh. 2.10 - Prob. 13ECh. 2.10 - Prob. 14ECh. 2.10 - Prob. 15ECh. 2.10 - Prob. 16ECh. 2.10 - Prob. 17ECh. 2.10 - Prob. 18ECh. 2.10 - Prob. 19ECh. 2.10 - Prob. 20ECh. 2.10 - Prob. 21ECh. 2.10 - 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