Calc2.37

From Exampleproblems

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.

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