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Find the derivative of f(x)\, with respect to x\,: f(x)=\ln(7x^{2}e^{x}\sin x)\,

At first glance, this problem looks to be very complicated but all we need are the laws of logarithms which allow us to split up products inside logarithms and turn them into sums of logarithms.

f(x)=\ln(7x^{2}e^{x}\sin x)=\ln 7+2\ln x+x\ln e+\ln(\sin x)\,

Now, we can take the derivatives of the seperate parts.

f'(x)={\frac  {2}{x}}+1+{\frac  {\cos x}{\sin x}}={\frac  {2}{x}}+1+\cot x\,

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