   Chapter 3.2, Problem 55E

Chapter
Section
Textbook Problem

Find R'(0), where R ( x ) = x − 3 x 3 + 5 x 5 1 + 3 x 3 + 6 x 6 + 9 x 9 Hint: Instead of finding R'(x) first, let f(x) be the numerator and g(x) the denominator of R(x) and compute R'(0) from f(0), f'(0), g(0), and g'(0).

To determine

To find: The value of R(0).

Explanation

Given:

The function R(x)=x3x3+5x51+3x3+6x6+9x9.

Derivative rule:

(1) Quotient Rule: If f1(x) and f2(x) are both differentiable, then

ddx[f1(x)f2(x)]=f2(x)ddx[f1(x)]f1(x)ddx[f2(x)][f2x]2

(2) Power rule: ddx(xn)=nxn1

(3) Sum rule: ddx(f+g)=ddx(f)+ddx(g)

(4) Difference rule: ddx(fg)=ddx(f)ddx(g)

(5) Constant multiple rule: ddx(cf)=cddx(f)

Calculation:

Let R(x)=f(x)g(x)

Where,f(x)=x3x3+5x5 and g(x)=1+3x3+6x6+9x9.

The derivative of the function R(x)=f(x)g(x) is R(x) which is obtained as follows,

R(x)=ddx(f(x)g(x))

Apply the Quotient rule (1) and simplify the terms,

R(x)=g(x)ddx(f(x))f(x)ddx(g(x))[g(x)]2=g(x)f(x)f(x)g(x)[g(x)]2

Substitute 0 for x in R(x),

R(x)=g(0)f(0)f(0)g(0)[g(0)]2 (1)

Obtain the value of f(0),

f(x)=ddx(x3x3+5x5)

Apply the derivative rule (3),(4) and (5),

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