Question

If \[ \frac{d}{dx} \left( \frac{(x+1)^2\sqrt{x-1}} {(x+4)^3e^x} \right) = f(x) \left[ \frac{2}{x+1} +\frac{1}{2(x-1)} -\frac{3}{x+4} -1 \right], \] then \(f(5)=\)

• A. \(\dfrac{72}{81}e^5\)
• B. \(\dfrac{7}{81e^5}\)
• C. \(\dfrac{8}{81e^5}\)
• D. \(e^5\)

Hint:

For expressions containing products, quotients and powers, logarithmic differentiation converts differentiation into a simple sum of terms.

Answer 1

The Correct Option is C

Step 1: Identify \(f(x)\).
Given \[ \frac{d}{dx} \left( \frac{(x+1)^2\sqrt{x-1}} {(x+4)^3e^x} \right) = f(x) \left[ \frac{2}{x+1} +\frac{1}{2(x-1)} -\frac{3}{x+4} -1 \right]. \] Using logarithmic differentiation, \[ \frac{d}{dx}(y) = y \left[ \frac{2}{x+1} +\frac{1}{2(x-1)} -\frac{3}{x+4} -1 \right]. \] Comparing with the given expression, \[ f(x)= \frac{(x+1)^2\sqrt{x-1}} {(x+4)^3e^x}. \]

Step 2: Evaluate \(f(5)\).
Substituting \(x=5\), \[ f(5) = \frac{(5+1)^2\sqrt{5-1}} {(5+4)^3e^5}. \] \[ = \frac{6^2\sqrt4} {9^3e^5}. \] \[ = \frac{36\times2} {729e^5}. \] \[ = \frac{72} {729e^5}. \] \[ = \frac{8} {81e^5}. \]

Step 3: Final conclusion.
Therefore, \[ \boxed{\frac{8}{81e^5}} \]