Question

If \[ \frac{d}{dx}\left(\frac{x^2+1}{(x^2+5)(x^2+9)}\right) = \frac{2x(x^2+1)}{(x^2+5)(x^2+9)} \left[ \frac{1}{f(x)}-\frac{1}{g(x)}-\frac{1}{h(x)} \right], \] then \(2h(x)-f(x)-g(x)=\)

• A. \(12\)
• B. \(16\)
• C. \(18\)
• D. \(20\)

Hint:

For differentiating rational functions written as products or quotients, logarithmic differentiation helps identify terms like \(\frac{1}{x^2+a}\) quickly.

Answer 1

The Correct Option is A

Step 1: Take logarithmic derivative of \(F(x) = \frac{x^2+1}{(x^2+5)(x^2+9)}\).
\(\ln F = \ln(x^2+1)-\ln(x^2+5)-\ln(x^2+9)\).
\[\frac{F'}{F} = \frac{2x}{x^2+1}-\frac{2x}{x^2+5}-\frac{2x}{x^2+9}\]

Step 2: Read off \(f, g, h\).
Comparing with \(2x F(x)\left[\frac{1}{f(x)}-\frac{1}{g(x)}-\frac{1}{h(x)}\right]\): \(f(x)=x^2+1,\; g(x)=x^2+5,\; h(x)=x^2+9\).

Step 3: Compute the required expression.
\[2h(x)-f(x)-g(x) = 2(x^2+9)-(x^2+1)-(x^2+5) = 2x^2+18-x^2-1-x^2-5 = 12\]
\[ \boxed{12} \]