Find the derivative of the following functions.
y = In |...

Question

Find the derivative of the following functions.

y = In |sin x|

Accepted Answer

Hello 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. Find the derivative of Y equals the natural log of the absolute value of cosine of X. Awesome. So it appears for this particular problem, we're ultimately trying to figure out what the derivative of this function of Y is. So we're trying to find the first derivative of this function of Y. So with that in mind, our first step is we need to note once again and recall that we are asked to evaluate the derivative of a logamithic function that also contains an absolute value function. So in order to do that, in order to solve for this, we need to first recall the definitions of the basic derivatives that we will use throughout the course of solving this particular problem. So we need to recall and note that D by DX, which is just a fancy way of saying we're taking the derivative with respect to X of the natural log of X is equal to 1 divided by X. And we also need to note that D by DX of cosine of X is equal to negatives of X. Awesome. So our second step now is we need to recall and recognize that the given logamithic function is a composite function that will require us to use the chain rule in order to solve for that derivative. So we need to note we can also neglect the absolute value function in the different differentiation step, but we will need to prove that during the differentiation of the two cases. And there's two cases because we need to consider the domain because this absolute value means that we're going to be dealing with the domain, so we're have to check out two separate conditions to make sure that the domain conditions are met. So first off, let us consider Y equals the natural log of the cosine of X with the domain that Let's consider the domain where a cosine of X is greater than 0, and when we do that, we could state that U is equal to cosine of X. And our third step. We can apply and use the chain rule now, so we could state that Y or this prime symbol indicates the first derivative of this function of Y, and it looks like an apostrophe that's in the power. So Y. is equal to D by DU of LN of U multiplied by D by DX of cosine of X. Fantastic, which is equal to 1 divided by U, multiplied by negative sign of X. Fantastic. So now our fourth step is we need to rewrite you as it approaches cosine of X. Fantastic. So with that in mind, We can go ahead and write that Y. is equal to 1 divided by a cosine of X multiplied by negative of X, which is equal to negatives of x divided by cosine of X. And our 5th step now is we need to recall and apply trigontric identities, and we need to note and recall that the fraction represents the tangent function, so we can go ahead and write that Y is equal to negative tangent of X. Now, for our 6th step, which is our final step, is we need to consider. That Y is equal to the natural log of negative cosine of X or the condition that cosine of X is less than 0, which once again, this is our second condition based off of the domain because of these absolute value signs or symbols, absolute value symbol. So, with that said, we need to write that Y is equal to the natural log of negative cosine of X. So let me take D by DX of the natural log of negative cosine of X, we will find that it's equal to -1 divided by the cosine of X. Multiplied by, so when you take -1 divided by the cosine of X, multiplied by negative cosine prime. of X, which this will be all equal to -1 divided by. Cosine of X multiplied by sin of X, which will be all equal to negative tangent. Of X. So therefore, without a doubt, since we get the same derivative, we could say in conclusion, the derivative of the function of Y will be Y is equal to negative tangent of X, and that is our final answer. Hooray, we did it. We saw for our final answer. Thank you so much for watching. Hopefully that helped, and I can't wait to see you on the next video. Bye.

Find the derivative of the following functions.
y = In |sin x|

Key Concepts

Derivative

Chain Rule

Logarithmic Differentiation

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