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How do you differentiate #sin(x^2)(cos(x^2))#?

Answer 1

Emma Aldridge

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

This problem can also be solved by directly applying the differentiation rules:

#d/dx sin(x^2) cos(x^2)#

First, use the product rule: #=sin(x^2) d/dx cos(x^2) + cos(x^2)d/dx sin(x^2) #

Then, each derivative can be solved using the chain rule: #=-sin(x^2) sin(x^2) d/dx x^2 + cos(x^2)cos(x^2)d/dx x^2 #

#=-2xsin^2(x^2) + 2xcos^2(x^2) #

At this point, we can simplify the expression by factoring out #2x# and applying the double angle identity #cos 2theta = cos^2 theta - sin^2 theta# :

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

#=2xcos(2x^2) #

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Answer from HIX Tutor

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.