How do you differentiate #y=e^(xcosx)#?

Answer 1

Christopher Allers

#y=e^(xcosx)#

Use the exponent rule for differentiating, and use the product rule. #dy/dx=[(x)(-sinx)+(1)(cosx)]*(e^(xcosx))#

#dy/dx=(cosx-xsinx)(e^(xcosx))#

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Answer 2

Ezekiel Alexander

To differentiate ( y = e^{x \cos x} ), you can use the chain rule and product rule. The chain rule states that if you have a function within a function, you differentiate the outer function first, and then multiply by the derivative of the inner function. The product rule states that if you have the product of two functions, the derivative is the first function times the derivative of the second function plus the second function times the derivative of the first function.

Let ( u = x \cos x ). Then ( \frac{du}{dx} = \cos x - x \sin x ).

Now, differentiate ( e^u ) with respect to ( u ), which is simply ( e^u ).

Therefore, ( \frac{dy}{dx} = e^{x \cos x} (\cos x - x \sin x) + e^{x \cos x} ).

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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.