Question
Differentiate the functions given in Exercise:
$(\text{x}+3)^2.(\text{x}+4)^3.(\text{x}+5)^4$
$(\text{x}+3)^2.(\text{x}+4)^3.(\text{x}+5)^4$
✓
Answer
Let $\text{y}=(\text{x}+3)^2.(\text{x}+4)^3.(\text{x}+5)^4\ \dots\text{(i)}$
Taking logs on both sides, we have
$\log\text{y}=2\log(\text{x}+3)+3\log(\text{x}+4)+4\log(\text{x}+5)^4$
$\therefore\ \frac{\text{d}}{\text{dx}}\log\text{y}=2\frac{\text{d}}{\text{dx}}\log(\text{x}+3)+3\frac{\text{d}}{\text{dx}}\log(\text{x}+4)+4\frac{\text{d}}{\text{dx}}\log(\text{x}+5)$
$\Rightarrow\ \frac{1}{\text{y}}\frac{\text{dy}}{\text{dx}}=2\frac{1}{\text{x}+3}\frac{\text{d}}{\text{dx}}(\text{x}+3)+3\frac{1}{\text{x}+4}\frac{\text{d}}{\text{dx}}(\text{x}+4)+4\frac{1}{\text{x}+5}\frac{\text{d}}{\text{dx}}(\text{x}+5)$
$\Rightarrow\ \frac{1}{\text{y}}\frac{\text{dy}}{\text{dx}}=\frac{2}{\text{x}+3}+\frac{3}{\text{x}+4}+\frac{4}{\text{x}+5}$
$\Rightarrow\ \frac{\text{dy}}{\text{dx}}=\text{y}\Big(\frac{2}{\text{x}+3}+\frac{3}{\text{x}+4}+\frac{4}{\text{x}+5}\Big)$
$\Rightarrow\ \frac{\text{dy}}{\text{dx}}=(\text{x}+3)^2(\text{x}+4)^3(\text{x}+5)^4\Big(\frac{2}{\text{x}+3}+\frac{3}{\text{x}+4}+\frac{4}{\text{x}+5}\Big)\ \text{[From eq.(i)]}$
Taking logs on both sides, we have
$\log\text{y}=2\log(\text{x}+3)+3\log(\text{x}+4)+4\log(\text{x}+5)^4$
$\therefore\ \frac{\text{d}}{\text{dx}}\log\text{y}=2\frac{\text{d}}{\text{dx}}\log(\text{x}+3)+3\frac{\text{d}}{\text{dx}}\log(\text{x}+4)+4\frac{\text{d}}{\text{dx}}\log(\text{x}+5)$
$\Rightarrow\ \frac{1}{\text{y}}\frac{\text{dy}}{\text{dx}}=2\frac{1}{\text{x}+3}\frac{\text{d}}{\text{dx}}(\text{x}+3)+3\frac{1}{\text{x}+4}\frac{\text{d}}{\text{dx}}(\text{x}+4)+4\frac{1}{\text{x}+5}\frac{\text{d}}{\text{dx}}(\text{x}+5)$
$\Rightarrow\ \frac{1}{\text{y}}\frac{\text{dy}}{\text{dx}}=\frac{2}{\text{x}+3}+\frac{3}{\text{x}+4}+\frac{4}{\text{x}+5}$
$\Rightarrow\ \frac{\text{dy}}{\text{dx}}=\text{y}\Big(\frac{2}{\text{x}+3}+\frac{3}{\text{x}+4}+\frac{4}{\text{x}+5}\Big)$
$\Rightarrow\ \frac{\text{dy}}{\text{dx}}=(\text{x}+3)^2(\text{x}+4)^3(\text{x}+5)^4\Big(\frac{2}{\text{x}+3}+\frac{3}{\text{x}+4}+\frac{4}{\text{x}+5}\Big)\ \text{[From eq.(i)]}$
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