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Question

If $$\displaystyle y= \frac {\sqrt[3]{1+3x}\sqrt[4]{1+4x}\sqrt[5]{1+5x}}{\sqrt[7]{1+7x}\sqrt[8]{1+8x}}$$, then $$y'(0)$$ is equal to


A
1
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B
1
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C
2
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D
Non existant
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Solution

The correct option is B $$1$$
$$\displaystyle y= \frac {\sqrt[3]{1+3x}\sqrt[4]{1+4x}\sqrt[5]{1+5x}}{\sqrt[7]{1+7x}\sqrt[8]{1+8x}}$$
Take log both sides,
$$\displaystyle \log y = \frac{1}{3}\log(1+3x)+\frac{1}{4}\log(1+4x)+\frac{1}{5}\log(1+5x)-\frac{1}{7}\log(1+7x)-\frac{1}{8}\log(1+8x)$$
Differentiating both side w.r.t $$x$$
$$\displaystyle
\frac {1}{y}\times \frac {dy}{dx}=\frac {1}{(1+3x)}+\frac
{1}{1+4x}+\frac {1}{1+5x}-\frac {1}{1+7x}-\frac {1}{1+8x}$$
at $$x=0, y=1$$,  
$$\therefore \displaystyle \frac {dy}{dx}=1+1+1-1-1=1$$

Mathematics

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