Calc2.37

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Find the derivative of f(x)\, with respect to x\,: f(x)={\frac  {{\frac  {\sin(5x)}{(x^{2}+1)^{2}}}}{\cos ^{3}(3x)-1}}\,

Here, we have a quotient rule where the numerator will involve another quotient rule and the chain rule will need be used several times.

f'(x)={\frac  {[\cos ^{3}(3x)-1]\left[{\frac  {(x^{2}+1)^{2}(5)\cos(5x)-\sin(5x)(2)(x^{2}+1)(2x)}{(x^{2}+1)^{4}}}\right]-{\frac  {\sin(5x)}{(x^{2}+1)^{2}}}[-9\cos ^{2}(3x)\sin(3x)]}{[\cos ^{3}(3x)-1]^{2}}}\,


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