Question

differentiate y=e^xsinxcos² x

Answer 1

$dy/dx = e^x * \frac{d}{dx} (sin\ x\ cos^2x)+sin\ x\ cos^2x * \frac{d}{dx}e^x\\ \\e^x* [sinx\ 2cos\ x(-sinx)+cos^2x\ cos\ x ] + sinx\ cos^2x\ e^x\\ \\e^x\ cos\ x [ 3cos^2x-2 +\frac{1}{2}sin2x ]\\\\= \frac{1}{2}e^x\ cos\ x [ 3cos2x -1+sin2x]$


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or, another way:

$dy/dx = \frac{d}{dx}e^x [ sinx - sin^3x]=\\ \\=e^x[ sin\ x - sin^3x + cos\ x - 3sin^2x\ Cos\ x ]\\ \\=e^x[ sinx+cosx-sin^2x( sinx+3cosx)]$