Understanding the Derivative Problem

The problem asks us to find the derivative of a given rational function and then use the resulting expression, which is in the form \(ax + bx^3\), to determine the correct relationship between the coefficients \(a\) and \(b\).

The given function is \(\frac{1+x^4+x^8}{1-x^2+x^4}\). To find the derivative, it's often helpful to simplify the expression first, if possible. Let's look at the numerator \(1+x^4+x^8\).

Simplifying the Rational Function

We can try to factor the numerator \(1+x^4+x^8\). This expression looks similar to the expansion of \((1+y+y^2)(1-y+y^2)\). Let's see if it relates to the denominator.

Consider the identity: \(1+u^2+u^4 = (1+u^2)^2 - u^2 = (1+u^2-u)(1+u^2+u)\).

Let \(u = x^2\). Then \(u^2 = x^4\) and \(u^4 = x^8\).

The numerator \(1+x^4+x^8\) can be written as \(1+(x^2)^2+(x^2)^4\), which doesn't directly fit the identity's form \(1+u^2+u^4\). Let's reconsider the factorization.

Consider the expression \(1+x^2+x^4\). Multiplying this by \((1-x^2+x^4)\):

\((1+x^2+x^4)(1-x^2+x^4)\)

Let \(A = 1+x^4\) and \(B = x^2\). This product is \((A+B)(A-B) = A^2 - B^2\).

So, \((1+x^4+x^2)(1+x^4-x^2) = (1+x^4)^2 - (x^2)^2 = (1 + 2x^4 + x^8) - x^4 = 1 + x^4 + x^8\).

Thus, the numerator \(1+x^4+x^8\) can be factored as \((1-x^2+x^4)(1+x^2+x^4)\).

Now, we can rewrite the original rational function:

\(\frac{1+x^4+x^8}{1-x^2+x^4} = \frac{(1-x^2+x^4)(1+x^2+x^4)}{1-x^2+x^4}\)

Assuming that \(1-x^2+x^4 \neq 0\), we can cancel the common factor:

\(\frac{1+x^4+x^8}{1-x^2+x^4} = 1+x^2+x^4\)

Finding the Derivative

Now we need to find the derivative of the simplified expression \(1+x^2+x^4\).

\(\frac{d}{d x}\left(1+x^2+x^4\right)\)

Using the power rule for differentiation, \(\frac{d}{dx}(x^n) = nx^{n-1}\), and the rule that the derivative of a constant is zero:

• \(\frac{d}{dx}(1) = 0\)
• \(\frac{d}{dx}(x^2) = 2x^{2-1} = 2x\)
• \(\frac{d}{dx}(x^4) = 4x^{4-1} = 4x^3\)

So, the derivative is:

\(\frac{d}{d x}\left(1+x^2+x^4\right) = 0 + 2x + 4x^3 = 2x + 4x^3\)

Comparing with \(ax + bx^3\)

We are given that \(\frac{d}{d x}\left(\frac{1+x^4+x^8}{1−x^2+x^4}\right) = ax + bx^3\).

From our calculation, the derivative is \(2x + 4x^3\).

By comparing the coefficients of the terms with the same power of \(x\), we have:

• The coefficient of \(x\) is \(a\) on the right side and \(2\) on the left side. So, \(a = 2\).
• The coefficient of \(x^3\) is \(b\) on the right side and \(4\) on the left side. So, \(b = 4\).

Checking the Options

Now we check which of the given options is correct using \(a=2\) and \(b=4\).

1. \(a = b\): Is \(2 = 4\)? No, this is false.

2. \(a = 2b\): Is \(2 = 2 \times 4\)? Is \(2 = 8\)? No, this is false.

3. \(a + b = 0\): Is \(2 + 4 = 0\)? Is \(6 = 0\)? No, this is false.

4. \(2a = b\): Is \(2 \times 2 = 4\)? Is \(4 = 4\)? Yes, this is true.

Therefore, the correct relationship between \(a\) and \(b\) is \(2a = b\).

Revision Table: Key Steps

Additional Information: Polynomial Factorization and Differentiation

Polynomial Factorization: The simplification relied on recognizing a specific factorization pattern. The identity \(1+x^{2n}+x^{4n} = (1-x^n+x^{2n})(1+x^n+x^{2n})\) is useful here. In our case, for the numerator \(1+x^4+x^8\), we can let \(n=2\). Then \(x^n = x^2\), \(x^{2n} = x^4\), and \(x^{4n} = x^8\). So, \(1+x^4+x^8 = (1-x^2+x^4)(1+x^2+x^4)\).

Basic Differentiation Rules:

• Power Rule: If \(f(x) = x^n\), then \(\frac{d}{dx}(f(x)) = nx^{n-1}\).
• Constant Rule: If \(f(x) = c\) (where \(c\) is a constant), then \(\frac{d}{dx}(f(x)) = 0\).
• Sum/Difference Rule: If \(h(x) = f(x) \pm g(x)\), then \(\frac{d}{dx}(h(x)) = \frac{d}{dx}(f(x)) \pm \frac{d}{dx}(g(x))\).

In this problem, we used the constant rule for the term \(1\) and the power rule for terms \(x^2\) and \(x^4\), combined using the sum rule.