63–74. Derivatives of logarithmic functions Calculate the deri...

Question

63–74. Derivatives of logarithmic functions Calculate the derivative of the following functions. In some cases, it is useful to use the properties of logarithms to simplify the functions before computing f'(x).

y = log₈ |tan x|

y = log₈ |tan x|

Accepted Answer

Below there, today we're going to solve the following practice problem together. So first off, let us read the problem and highlight all the key pieces of information that we need to use in order to solve this problem. Calculate the derivative of the function Y equals logba 3 of the absolute value of sin of X. Awesome. So it appears for this particular problem, our final answer that we're ultimately trying to solve for is we're trying to figure out what the derivative of this function of Y is. So we're trying to figure out what Y is. That is our final answer we're ultimately trying to solve for the derivative of this function of Y. So with that in mind, our first step that we need to take, so first off, we need to recall and note that the derivative of a logametric function. Meaning, so we need to recall and note that for our log rule, Y equals log base A of X will be for, so if we take the derivative, which Y denotes the first derivative of Y, so that's what the the asterisks and the power means. So when you have an asterisk as a power, Then that means it's a derivative. So Y is the first derivative. And so as we should recall from our log rule, when we do loga mythic function derivatives, that Y will be equal to 1 divided by X multiplied by the natural log of A. So once again, Y will be 1 divided by X multiplied by the natural log of A. Fantastic. So now, we need to also note and recall for this particular problem since we're dealing with a composite function because Y is a composite function, we need to apply this elementary definition, so we need to apply our log rule, and we also need to use the chain rule and Because we need to use the log rule and the chain rule, and this is all due to the fact that we're dealing with a composite function because Y, once again, is a composite function composite function. So, with that said, we now, now that we know we note that we need to recall and use the chain rule, and we need to use this log rule. Our second step in order to solve This problem is we need to recall a note that the absolute value allows Y equals of X to obtain both positive and negative values. However, we have to recall that, and this is very important, and I'll write it in blue, we have to recall that. Log Prime Of a multiplied by X is equal to log a of the absolute value of X, and the proof for this particular expression is that if we state that log base A of the absolute value of X is equal to curly bracket of log base A of X. X is greater than 0, and log base A of negative X. Is X is less than 0, and we can also state that log. So log prime space A. Of X or and it's more specifically log prime base A of the absolute value of X is equal to curly bracket. Of 1 divided by X multiplied by the natural log of A, where X is greater than 0 and -1 divided by X multiplied by the natural log of A, where X is less than 0. Fantastic. And that once again is our proof. So now, our next step, we need to recall a note that by taking the same X value with a negative sign in the second piece-wise function will still yield a positive product. And we need to recall a note that the same expression as given in the first part of our piece-wise function. So, once again, this will be the same expression that is given to us in the first part of our piece-wise function. So with that said, we can now start our 3rd step to help us solve for this particular problem. So, step 3, we need to recall and note that we need to evaluate the derivative of, and we're recalling how to evaluate the derivative of derivative of Y equals log base 3. Of sine of X. And in order to do that, we need to apply the chain rule, and we also need to recall that sine of X is equal to cosine of X. So you could put a parentheses around the X. But you can also leave the parentheses off, it's whatever notation makes the most sense to you, but we have to note that this is stating that sine of X is equal to cosine of X. So with that in mind, we can go ahead and write that Y. is equal to one divided by Sign of X multiplied by the natural log of 3, multiplied by D by DX of sine of X. So, when we simplify, we will find that Y is equal to 1 divided by the natural log of 3 multiplied by sin of X, all multiplied by cosine of X. Awesome. So now, our next step. And let's make a little barrier here, since we're going on to step 4 now. Our next step now is we need to recall a note that we need to simplify our expression now using trigontric identities. So we're recalling and using our trig identities, we can go ahead and write that. is equal to 1 divided by the natural log of 3 multiplied by cosine of x divided by sin of X, and when we simplify, we will find that Y is equal to cotangent of X divided by the natural log of 3. And without a doubt, that will be our final answer for the derivative of Y of our function of Y. So once again, our final answer, which is the derivative of this function of Y, will be Y is equal to cotangent of X divided by the natural log of 3. Thank you so much for watching. Hopefully that helped, and I can't wait to see you in the next video. Bye.

63–74. Derivatives of logarithmic functions Calculate the derivative of the following functions. In some cases, it is useful to use the properties of logarithms to simplify the functions before computing f'(x).y = log₈ |tan x|

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63–74. Derivatives of logarithmic functions Calculate the derivative of the following functions. In some cases, it is useful to use the properties of logarithms to simplify the functions before computing f'(x).

y = log₈ |tan x|