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: Use logarithmic differentiation form.
Let \[ F(x)=\frac{x^2+1}{(x^2+5)(x^2+9)}. \] Then, \[ F'(x)=F(x)\left[ \frac{2x}{x^2+1} -\frac{2x}{x^2+5} -\frac{2x}{x^2+9} \right]. \]

Step 2: Take common factor.
\[ F'(x)= \frac{x^2+1}{(x^2+5)(x^2+9)} \cdot 2x \left[ \frac{1}{x^2+1} -\frac{1}{x^2+5} -\frac{1}{x^2+9} \right]. \] So, \[ F'(x)= \frac{2x(x^2+1)}{(x^2+5)(x^2+9)} \left[ \frac{1}{x^2+1} -\frac{1}{x^2+5} -\frac{1}{x^2+9} \right]. \]

Step 3: Compare with the given expression.
Comparing with \[ \frac{2x(x^2+1)}{(x^2+5)(x^2+9)} \left[ \frac{1}{f(x)}-\frac{1}{g(x)}-\frac{1}{h(x)} \right], \] we get \[ f(x)=x^2+1, \] \[ g(x)=x^2+5, \] and \[ h(x)=x^2+9. \]

Step 4: Find \(2h(x)-f(x)-g(x)\).
\[ 2h(x)-f(x)-g(x) = 2(x^2+9)-(x^2+1)-(x^2+5) \] \[ =2x^2+18-x^2-1-x^2-5 \] \[ =12. \]

Step 5: Final conclusion.
Therefore, \[ \boxed{12} \]