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Find the derivative of $y$ with respect to $x, t,$ or $\theta,$ as appropriate.\begin{equation}y=e^{\theta}(\sin \theta+\cos \theta)\end{equation}

$y^{\prime}=2 e^{\theta} \cos \theta$

Calculus 1 / AB

Chapter 7

Transcendental Functions

Section 3

Exponential Functions

Functions

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we want to find the derivative of y with respect to X, to data as appropriate. So the first thing we should do is go about finding what we should be taking this with respect to so notice. The only variable we have floating around here, this data. So it makes sense for us to take the derivative of this function with respect to data. Because otherwise we take with respect, expertise. We just end up with zero. And a derivative of zero is not very interesting everywhere. All right, so let's go ahead and take our driven with prospective data. So we get d by detaining why is equal to D by data of e later multiplied by scientific data plus co sign. All right now the left hand side here. So when I take the drove over my with prospective data, we just get wide prime or delight by data and then for us to take the derivative of what we have, you're gonna need to use the plot of cool somebody. Go ahead. Right. So we need product. Remember, product fool says we keep one of them the same and then take the derivative of the other So do you buy of sign data co sign Beta, and then we add that to the zoo Just switched places. So science Data Plus co signed data multiplied by the derivative with prospective data of you Now remember, derivative Oh, you to the ex with respect to X is just going to be the ex. So we'll need this here now to take the derivative. Oh, signed data plus go someday to remember that I need to distribute being derivative and then take the derivatives individually. So the derivative of science data is co sign Beta and then the derivative of CO signed data is minus sign beta. And then over here we just have signed beta plus co sign data and using that identity right above to take the derivative of Eaton's data. It is going to beat me to the data. So you just left with you to the fate of their All right. So now if I were to distribute to the data into both of these, notice that we end up with E to date a minus sign data in this first term, and in the second term, we get me to the sign fatal so these two terms cancel out. And then we would just be left with e beta high school science data plus e fate of coastline data so we can simplify that down too. Two e to the beta co sign data for our derivative of dysfunction with respect to

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