Question

find the differential dy where y=sin^2(3x)sqrt(x^3+7)

Answer 1

\[\begin{align*}dy&=\frac{d}{dx}\left[\sin^23x\sqrt{x^3+7}\right]\,dx\\\\
&=\left(\frac{d}{dx}\left[\sin^23x\right]\sqrt{x^3+7}+\sin^23x\frac{d}{dx}\left[\sqrt{x^3+7}\right]\right)\,dx\\\\
&=\left(2\sin3x\frac{d}{dx}\left[\sin3x\right]\sqrt{x^3+7}+\frac{\sin^23x}{2\sqrt{x^3+7}}\frac{d}{dx}\left[x^3+7\right]\right)\,dx\\\\
&=\left(2\sin3x\cos3x\frac{d}{dx}\left[3x\right]\sqrt{x^3+7}+\frac{3x^2\sin^23x}{2\sqrt{x^3+7}}\right)\,dx\\\\
&=\left(6\sin3x\cos3x\sqrt{x^3+7}+\frac{3x^2\sin^23x}{2\sqrt{x^3+7}}\right)\,dx\\\\
&=\left(3\sin6x\sqrt{x^3+7}+\frac{3x^2\sin^23x}{2\sqrt{x^3+7}}\right)\,dx
\end{align*}\]

Answer 2

thank you