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Chapter 6: Problem 14

If \(\frac{x^{4}+x^{2}+1}{x^{2}-x+1}\) such that \(\frac{d y}{d x}=a x+b\), find the, value of \(a+b+10\)

Short Answer

Expert verified
The value of \(a+b+10\) is 9.

Step by step solution

01

Identify the Function and Derive it

The function is given as \(f(x) = \frac{x^{4} + x^{2} + 1}{x^{2} - x + 1}\). We must derive it using the quotient rule. The quotient rule states that \((\frac{f}{g})' = \frac{f'g - f g'}{g^2}\) where \(f\) and \(g\) are functions. Let \(f(x) = x^{4} + x^{2} + 1\) and \(g(x) = x^{2} - x + 1\). First, compute the derivatives \(f'(x)\) and \(g'(x)\). We get \(f'(x)= 4x^{3} + 2x\) and \(g'(x)= 2x - 1\). Then, apply the quotient rule, which gives us the derivative of the function as \(f'(x) = \frac{(4x^{3} + 2x)(x^{2} - x + 1) - (x^{4} + x^{2} + 1)(2x - 1)}{(x^{2} - x + 1)^2}\).
02

Simplify the Derived Function

To simplify, we must expand the terms in the numerator and then collect similar terms. After simplification, we get \(f'(x) = 2x - 3\).
03

Compare the Derived Function with General Form

The derived function \(f'(x) = 2x - 3\) has the general form \(ax+b\). So comparing these two gives \(a = 2\) and \(b = -3\).
04

Computing the Value of \(a + b + 10\)

Substitute the values of \(a\) and \(b\) into \(a + b + 10\). This gives us the required value as \(2 - 3 + 10 = 9\).

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