A) \[{{\left( 2-x \right)}^{3}}{{\left( 3+2x \right)}^{5}}\left[ \frac{15}{3+2x}-\frac{8}{2-x} \right]\]
B) \[{{\left( 2-x \right)}^{3}}{{\left( 3+2x \right)}^{5}}\left[ \frac{15}{3+2x}+\frac{3}{2-x} \right]\]
C) \[{{\left( 2-x \right)}^{3}}{{\left( 3+2x \right)}^{5}}\left[ \frac{10}{3+2x}-\frac{3}{2-x} \right]\]
D) \[{{\left( 2-x \right)}^{3}}{{\left( 3+2x \right)}^{5}}\left[ \frac{10}{3+2x}+\frac{3}{2-x} \right]\]
Correct Answer: C
Solution :
\[y={{\left( 2-x \right)}^{3}}{{\left( 3+2x \right)}^{5}}\] \[\Rightarrow \,\log \,y=\log {{\left( 2-x \right)}^{3}}+\log {{\left( 3+2x \right)}^{5}}\] \[=3\log \,\left( 2-x \right)+5\,\log \left( 3+2x \right)\] \[\Rightarrow \frac{1}{y}\frac{dy}{dx}=\frac{3\times \left( -1 \right)}{2-x}+\frac{5}{3+2x}\times \left( 2 \right)\] \[\Rightarrow \,\frac{dy}{dx}={{\left( 2-x \right)}^{3}}{{\left( 3+2x \right)}^{5}}\left[ \frac{10}{3+2x}-\frac{3}{2-x} \right]\]
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