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  • Differentiate each of the following w.r.t. x (x + 1)^2 (x + 2)^3 (x + 3)^4

Differentiate each of the following w.r.t. x  (x + 1)^2 (x + 2)^3 (x + 3)^4

Answers (1)

We have given,  (x + 1)^2 (x + 2)^3 (x + 3)^4

Let us Assume,  y=(x + 1)^2 (x + 2)^3 (x + 3)^4

Taking log both side

\\Log y=log [(x + 1)\textsuperscript{2} (x + 2)\textsuperscript{3} (x + 3)\textsuperscript{4}] \\Log y=log(x + 1)\textsuperscript{2} +log(x + 2)\textsuperscript{3} +log(x + 3)\textsuperscript{4} \\Log y=2log(x + 1) +3log(x + 2) +4log(x + 3)\\

Differentiate w.r.t x

\\ \frac{1}{y} \cdot \frac{d y}{d x}=\frac{2}{x+1}+\frac{3}{x+2}+\frac{4}{(x+3)} \\ \frac{d y}{d x}=y\left[\frac{2}{x+1}+\frac{3}{x+2}+\frac{4}{(x+3)}\right] \\ \frac{d y}{d x}=(x+1)^{2} \cdot(x+2)^{3} \cdot(x+3)^{4}\left[\frac{2}{x+1}+\frac{3}{x+2}+\frac{4}{x+3}\right]

\\ =(x+1)^{2} \cdot(x+2)^{3} \cdot(x +3)^{4}\left[\frac{2(x+2)(x+3)+3(x+1)(x+3)+4(x+1)(x+2)}{(x+1)(x+2)(x+3)}\right] \\ =(x+1)(x+2)^{2}(x+3)^{3}\left[9 x^{2}+34 x+29\right]

 

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infoexpert22

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