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: Recognise f(x) from logarithmic differentiation structure.
If \(y=\frac{(x+1)^2\sqrt{x-1}}{(x+4)^3e^x}\), then \(\frac{dy}{dx}=y\left[\frac{2}{x+1}+\frac{1}{2(x-1)}-\frac{3}{x+4}-1\right]\). So \(f(x)=y=\frac{(x+1)^2\sqrt{x-1}}{(x+4)^3e^x}\).

Step 2: Evaluate f(5).
\(f(5)=\frac{6^2\cdot\sqrt{4}}{9^3\cdot e^5}=\frac{36\cdot2}{729e^5}=\frac{72}{729e^5}=\frac{8}{81e^5}\). \[\boxed{\dfrac{8}{81e^5}}\]