I am not getting this 48 in the differentiation of $\frac{3}{5(2x+3)}$.

Can you please verify once??

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Let $f(x) = \dfrac{x}{(x-1)(2x+3)}$, where $x>1$. Then the $4^{th}$ derivative of $f, \: f^{(4)} (x)$ is equal to

- $- \frac{24}{5} \bigg[ \frac{1}{(x-1)^5} – \frac{48}{(2x+3)^5} \bigg]$
- $\frac{24}{5} \bigg[ – \frac{1}{(x-1)^5} + \frac{48}{(2x-3)^5} \bigg]$
- $\frac{24}{5} \bigg[ \frac{1}{(x-1)^5} + \frac{48}{(2x+3)^5} \bigg]$
- $\frac{64}{5} \bigg[ \frac{1}{(x-1)^5} + \frac{48}{(2x+3)^5} \bigg]$

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$\underline{\textbf{Answer:}}\Rightarrow \mathbf C$

$\underline{\textbf{Explanation:}}\Rightarrow$

Using partial differentiation:

$\frac{x}{(x-1)(2x+3)} = \frac{\mathrm A}{x-1} + \frac{\mathrm B}{(2x+3)}$

$\Rightarrow \mathrm x = \mathrm A(2x+3) + \mathrm B(x-1)$

When $x = 1$, then $1 = 5\mathrm A \Rightarrow \mathrm A = \frac{1}{5}$

when $x = -\frac{3}{2}, \text {then}\; -\frac{3}{2} = -\frac{5\mathrm B}{2} \Rightarrow \mathrm B = \frac{3}{5}$

So, $f(x) = \frac{x}{(x-1)(2x+3)} = \frac{1}{(x-1)}+\frac{3}{5(3x+3)}$

Now, $f(x) = \frac{1}{5(x-1)} + \frac{3}{5(2x+3)}\\ f'(x) =\frac{-1}{5(x-1)^2} - \frac{6}{5(2x+3)^2}\\ f''(x) = \frac{2}{5(x-1)^3}+ + \frac{24}{5(2x+3)^3}\\ f'''(x) = \frac{-6}{5(x-1)^4} - \frac{72\times2}{5(2x+3)^4} \\f''''(x) = \frac{24}{5(x-1)^5} + \frac{72\times2\times8}{5(2x+3)^5} \\f''''(x) = \frac{24}{5}\bigg[\frac{1}{(x-1)^5} + \frac{48}{5(2x+3)^5}\bigg ]$

$\therefore \mathrm {\mathbf C}$ is the correct option.

$\underline{\textbf{Explanation:}}\Rightarrow$

Using partial differentiation:

$\frac{x}{(x-1)(2x+3)} = \frac{\mathrm A}{x-1} + \frac{\mathrm B}{(2x+3)}$

$\Rightarrow \mathrm x = \mathrm A(2x+3) + \mathrm B(x-1)$

When $x = 1$, then $1 = 5\mathrm A \Rightarrow \mathrm A = \frac{1}{5}$

when $x = -\frac{3}{2}, \text {then}\; -\frac{3}{2} = -\frac{5\mathrm B}{2} \Rightarrow \mathrm B = \frac{3}{5}$

So, $f(x) = \frac{x}{(x-1)(2x+3)} = \frac{1}{(x-1)}+\frac{3}{5(3x+3)}$

Now, $f(x) = \frac{1}{5(x-1)} + \frac{3}{5(2x+3)}\\ f'(x) =\frac{-1}{5(x-1)^2} - \frac{6}{5(2x+3)^2}\\ f''(x) = \frac{2}{5(x-1)^3}+ + \frac{24}{5(2x+3)^3}\\ f'''(x) = \frac{-6}{5(x-1)^4} - \frac{72\times2}{5(2x+3)^4} \\f''''(x) = \frac{24}{5(x-1)^5} + \frac{72\times2\times8}{5(2x+3)^5} \\f''''(x) = \frac{24}{5}\bigg[\frac{1}{(x-1)^5} + \frac{48}{5(2x+3)^5}\bigg ]$

$\therefore \mathrm {\mathbf C}$ is the correct option.