Question

Find f'(x) of [sinx².sin²x]/(1 + sinx)

Answer 1

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$\frac{d}{dx}\left(\frac{\left[sinx².\:sin²x\right]}{\left[1+sinx\right]}\right)$

Apply the quotient rule

$\left(\frac{f}{g}\right)'=\frac{ f' \cdot g-g'\cdot f}{g^2}$

$=\frac{\frac{d}{dx}\left(\sin \left(x^2\right)\sin ^2\left(x\right)\right)\left(1+\sin \left(x\right)\right)-\frac{d}{dx}\left(1+\sin \left(x\right)\right)\sin \left(x^2\right)\sin ^2\left(x\right)}{\left(1+\sin \left(x\right)\right)^2}$

$= \frac{\left(\cos \left(x^2\right)\cdot \:2x\sin ^2\left(x\right)+\sin \left(2x\right)\sin \left(x^2\right)\right)\left(1+\sin \left(x\right)\right)-\cos \left(x\right)\sin \left(x^2\right)\sin ^2\left(x\right)}{\left(1+\sin \left(x\right)\right)^2}$