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Problem 8

In Exercises $1-20,$ find the derivative. $$y=x^…

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Problem 7

In Exercises $1-20,$ find the derivative.
$$y=\ln (\sin t+1)$$

Answer

$y^{\prime}=\frac{\cos t}{\sin t+1}$


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Video Transcript

All right, So we're going through this again. We're gonna look at a walkthrough for questions. Seven on the 3.9 exercises. So again, with this one, make sure you are familiar with your basic differentiation. Rules for Constance addition, subtraction, multiplication division as well as multiple functions and powers. Then adding on the new stuff, you're looking at your exponential along with MK functions. So all of your different derivatives for that how they apply, if I've got that with functions added, and then going through your trig functions as well from your previous chapter. So derivatives of your trip functions as well as understanding what those other functions So all of your inverse or your co seeking you're seeking in your co tangent, which are your rational functions for a sign coast in town. Um, co tangent is also won over 10 which also works out to the flipped or inverse of tan, which is coast over sign. So looking at this problem here, we've got the natural log of signed T plus one. So you've got a couple things at play here. We've got the chain rule we're gonna have to use because we've got multiple functions that are embedded into each other. So we've got our F function, which would be like our natural log. Then we have our G functional GBR signed T plus one. And looking at the derivative of that, we'd start with the derivative of the natural log and then work our way into the brackets and look at our derivative for r sine function. We also are dealing with our natural log here. So looking at the derivative of a natural log which works out to one over X and then we're also looking at the derivative of a trig function. So the derivative of sign becomes co sign. We're going to use those three properties here, tow walk through this problem and solve it. So starting with the chain rule, we're gonna take the derivative of our first function, which is their natural log. That'd be one over X in this case are X is our G of X said B R sine of t plus one. So there is our f prime of G, and now we're gonna work into the brackets. So we're going to deal with the sine function now. This is an addition problem where there's a constant, so we only actually have to take the derivative of the sine function, which would leave us with co sign, and then our plus one becomes zero because it's a constant. So you're also dealing with your constant rule if you want to think about that as well, where the derivative of a constant is zero. So looking at this, you could clean it up a little bit more by just multiplying that in so you could write. This is if you want as coasts over signed T plus one, and that would be as far as you can go with the derivative in this case.

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