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How do i differentiate #xe^(xy)cos(2x)# with respect to x??

Answer 1

Cameron Alexander

This is actually a multiplication rule within a multiplication rule.

I'm going to evaluate it as:

#(xe^xy)(cos(2x))#

Remember the rule for multiplication:

first(derivative of the 2nd) + second(derivative of the first)

You also need to remember that you must take the derivative of the "inside" of the cos.

#(xe^xy)(-2sin(2x))+(cos(2x))(x(e^(xy)y)+e^(xy))#

#-2xe^(xy)sin(2x)+xye^(xy)cos(2x)+e^(xy)cos(2x)#

We can factor out #e^(xy)#

#e^(xy)(-2xsin(2x)+xycos(2x)+cos(2x))#

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Answer 2

Cameron Alexander

To differentiate ( xe^{xy}\cos(2x) ) with respect to ( x ), you can use the product rule followed by the chain rule:

[ \frac{d}{dx}\left(xe^{xy}\cos(2x)\right) = e^{xy}\cos(2x) + xe^{xy}(-2\sin(2x) + y\cos(2x)) ]

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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.