How do you differentiate #f(x)= sin(x^2)cos(x^2)# using the product rule?

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Answer 1 · 2015-12-05T00:48:58

#f'(x)=2xcos(2x^2)#

#f'(x)=cos(x^2)d/dx[sin(x^2)]+sin(x^2)d/dx[cos(x^2)]#

#d/dx[sin(x^2)]=cos(x^2)d/dx[x^2]=2xcos(x^2)#

#d/dx[cos(x^2)]=-sin(x^2)d/dx[x^2]=-2xsin(x^2)#

Plug back in.

#f'(x)=2xcos^2(x^2)-2xsin^2(x^2)#

#f'(x)=2x(cos^2(x^2)-sin^2(x^2))#

Note that #cos^2(a)-sin^2(a)=cos(2a)#.

#f'(x)=2xcos(2x^2)#