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Part B for X is not equal to zero express f prime of X is a piecewise defined function find the value of x for which f prime of x is equal to negative three so the first thing you might be wondering is why did we even have to take out X is equal to zero from there or why do why is the derivative not going to be defined there and that's just because you're going to see that the derivative is going to be something different when we approach X is equal to zero from the left vs. when we approach X is equal to zero from the right and that's why they just took it out of there for us but let's just figure out what that derivative is for all the other values of X so f prime of X is equal to so for X is less than zero we're going to take the derivative of this first case so the derivative of 1 is just 0 the derivative of negative 2 sine of X will derivative sine of X is just cosine of X it's just going to be negative 2 cosine of X negative 2 cosine of X 4 for X is less than 0 and then for X is a greater than 0 I'll do this in another color I'll do it in orange we have this case right over here and we'll just do the chain rule derivative of negative 4x with respect to X is negative 4 and derivative of e to the negative 4x with respect to e with the derivative of e to the negative 4x with respect to negative 4 X is just e to the negative 4x e to the negative 4x sometimes you could say this is derivative of the inside times the derivative of the outside with respect to the inside so either way it's negative 4 e to the negative 4x 4x is greater than 0 so we did the first part we expressed F prime of X as a piecewise defined function it's not we didn't define the derivative actually I forgot a parenthesis here we did interrupt we didn't define the derivative when X is equal to 0 because it's actually not going to be defined there now let's do the second part find the value of x for which F prime of X is equal to negative 3 and so if this wasn't piecewise defined you just very simply just say look you know f prime of X is equal to negative 3 you would take whatever f prime of X is equal to and you do some algebra to solve for it but here you're like well which - I didn't use I don't know if the X that gets us to negative 3 is gonna be less than 0 or I don't know if it's gonna be greater than 0 so I don't know which case to use and one thing that they realize is is to look at these functions a little bit and realize that cosine of X is a bounded function cosine of X can only go between positive 1 and negative 1 so negative 2 cosine of X can only go between positive 2 and negative 2 so this can only go between positive 2 and negative 2 so it can never get to negative 3 so if anything's ever going to get to negative 3 it's going to have to be this part of the derivative or this part of the derivative definition so it's going to have to be this thing right over here and hopefully there's some values of X greater than 0 where this thing right over here is equal to negative 3 so let's try it out negative 4 e to the negative 4x needs to be equal to negative 3 we can divide both sides by negative 4 we get e to the negative 4x is equal to negative 3 fourths divided by negative 4 is 3/4 we can take the natural log of both sides and we will get negative 4 X is equal to the natural log of 3/4 natural log of 3/4 and just to be clear what I did here I mean you literally could put the natural log here natural log there and you could put the natural log there as well to see that step the natural this is saying what power what power would have to raise e to to get e to the negative 4x well obviously I need to raise e to the just the negative 4x spot over there so this this power is negative 4x and then we just took the natural log of the right-hand side as well and then to solve for x we can divide both sides by negative 4 so you get X is equal to or we could say we could multiply both sides by negative 1/4 either way negative 1/4 natural log of 3/4 and what we need to do is verify that this X so we used this case right over here we used that case but we have to make sure that this X that we that we can use this case that this X is greater than 0 and we might be tempted right when we look at this is a way way this looks like a negative number but we have to remind ourselves the natural log of 3/4 since 3/4 is less than e the natural log of 3/4 is going to be a negative number is going to be e to some negative exponent so since this is negative and and this part right over here is negative so that is also negative you have a negative times a negative so this right over here is going to be positive so this is a positive value right over here so you would use you would use you would use this case right over here so that's our answer X is equal to negative 1/4 natural log of 3/4 or the derivative we could write F prime of negative 1/4 times the natural log of 3/4 is equal to is equal to negative 3 and we're done
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