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

Question

Find the derivative of the following functions.

y = In |x²-1|

Accepted Answer

Hello there. Today we're gonna 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 the function Y equals the natural log of the absolute value of X to the power of 3 minus 3 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 in order to do that, our first step before we do any differentiation, so prior to differentiation, we need to figure out and define our domain for this particular problem. So now we need to focus on figuring out the domain of this particular problem. So, we need to focus our attention on the absolute value of X to the power of 3 minus 3 X. So, the absolute value, once again, the absolute value of X to the power of 3 minus 3 X, and this has to be greater than 0. So that means that the absolute value of the function will consist of two parts. So we could say that Y is equal to Curly bracket of the natural log of X to the power of 3 minus 3 X where x to the power of 3 minus 3X is greater than 0, and the natural log of 3x minus X to the power of 3, where X 3 minus 3 X is less than 0. Awesome. So that is our absolute value function consisting of two parts, and this will help us define our domain once again. So now our second step, now that we've figured out our domain, we now need to differentiate the first part. So let us state for this particular case that you is equal to x to the power of 3 minus 3 X. And if we state that you is that value, we could go ahead and write that Y is equal to the natural log of u of X. Fantastic. And because Y equals the natural log of UX is a composite function. And because it's a composite function, we need to recall and apply the chain rule. So that will be our third step is applying the chain rule. So we need to state that Y, where this apostrophe symbol and the power that denotes the symbol for prime, and prime means so this means that we're taking the first derivative of the function of Y. So Y is equal to the natural log of u. Subscript U to the prime, so you, so this is the natural log of U subscript U, multiplied by UX. Awesome. So now our 4th step that we need to take is we need to evaluate the first derivative by recalling and using the table of basic derivatives. So when we do that, we will be able to write that the natural log of you, so the natural log so the natural log of you baseu prime. I, I said base, but it's subscript U, specifically subscript U, will be equal to 1 divided by you, which is going to be equal to 1 divided by X to the power of 3 minus 3 X. Fantastic. So now, Our next step, so our 5th step is we need to evaluate the 2nd derivative, and we need to recall that. A multiplied by X, the power of N is equal to a multiplied by N, multiplied by X to the power of N minus 1. So therefore, we go ahead and write that X to the power of 3 minus 3X subscript X is equal to 3. X2 minus 3. And now we can start solving for our 6th step, which for step 6, we need to substitute into the expression. So we need to take our values that we just found together and substitute them into our expression that is given to us in step 3. So when we do that, we will find out why. is equal to one divided by X to the power of 3 minus 3 X multiplied by 3 X2 minus 3. Which we could go ahead and simplify a little bit and write that Y is equal to 3 X2 minus 3 divided by X of 3. So when you take X to the power of 3. And we need to subtract 3 X. fantastic. And now for our final step for step 7, we need to differentiate the second part using the same steps, and we need to let you equal 3X minus X to the power of 3. So when we do that, we will find that Y is equal to 1 divided by 3X minus X to the power of 3, multiplied by D by DX 3X minus X to the power of 3. Which we can simplify to Y equals 3 minus 3 X 2. Divided by 3 X minus X to the power of 3. Fantastic. So that will mean that when we simplify one last time, Y will be equal to 3 X2 minus 3 divided by X of 3. So when you take this to X of 3. Fantastic. And then we need to subtract 3 X. Fantastic, and that will be our final answer. Hooray, we did it, and let's move this up above so we could see it a little bit better. So, our final answer once again will be Y is equal to 3 x 2 minus 3, all divided by X of 3. So let's write that a little clearer. So X to the power of 3. -3 X. Hooray, we did it. And that is our final answer once again for our first derivative for the function of Y. Hooray. And so this problem is very easy to solve for as long as you have a strong understanding of the different rules you have to apply in order to solve this particular derivative. So you need to understand how to use the chain rule and the power rule. Those are two very important rules to have memorized and know how to use and manipulate. And you also need to understand the logarithm. You need to understand natural logs and how to differentiate those too. And once you do, solving a problem like this is fairly simple and easy to do. Thank you so much for watching. Hopefully that helped, and I can't wait to see you in the next video. Bye.

Find the derivative of the following functions.
y = In |x²-1|

Key Concepts

Derivative

Natural Logarithm

Absolute Value Function

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