Find the derivative of the following functions.
y = In(c...

Question

Find the derivative of the following functions.

y = In(cos² 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 the function Y equals the natural log of sin squared of X. Awesome. So it appears for this particular problem, we're ultimately trying to figure out what the function of Y's derivative is. So we're trying to find the first derivative of this function of Y. So now that we know that we're trying to find the derivative of this function of Y. Our first step that we need to take is we need to recall a note when we look at this function by itself, we need to note that this given function contains an outer logamithic function and an inner composite composite function. And because it has an outer logamithic function and it contains an inner composite function, therefore, as a result of this fact, we will need to apply. So we're gonna have to recall and apply the chain rule twice. So, let us, as a result, let us state that U is equal to sin squad of X. So if U is equal to square of X, therefore, as a result, Y will equal the natural log of u. Awesome. So our second step now is we need to recall and apply the chain rule. We need to recall that the natural log prime of X is equal to 1 divided by X from the from our table of basic derivatives that we should recall, and as a result, we could go ahead and easily write that Y is equal to the natural log prime. Of you, which is equal to 1 divided by you, multiplied by U, which we could go ahead and write that Y is equal to 1 divided by the sine squared of X. So sine squared of X. And we need to multiply this value by Sin squared of X. Fantastic. So now for our third step is we need to recall and apply the chain rule for a 2nd time since sine squared of X is a composite function. We also need to recall a note that X to the power of A. is equal to a multiplied by X to the power of a minus 1. So when we evaluate this derivative, we will get that. Sine square of X, so we're gonna take sine squad of X is equal to 2 multiplied by the power of 2 minus 1 of X multiplied by Sign of X. is going to be equal to 2 multiplied by the sign of X, multiplied by, and we're running out of room, so I'm gonna move this down below. So this will mean that it's equal to 2 multiplied by sin x. Awesome. So it would be 2 multiplied by sin x multiplied by sin x. fantastic. So, with that said, we are now gonna need to switch gears here just a little bit, or really briefly. And our next step now for step 4, we need to recall and use our table of basic derivatives and let us state that. Sineme of X is equal to cosine of x as we should recall, and we need to then calculate the result from step 3, and when we do that, we will find that Sign Squared of X, so sin squared of X is equal to 2 multiplied by sin x multiplied by cosine of X. Fantastic. So now, our fifth step is we need to substitute the result from step 4 into our expression that we found in step 2. So we need to write that Y. Which once again, this prime symbol is the first derivative. That's what the prime symbol means. It's the first derivative, and it looks like an apostrophe in the power. So this is the first derivative of the function of Y. is equal to 1 divided by sin square of X, multiplied by 2 multiplied by sin. Of X multiplied by cosine of X, which is equal to 2 multiplied by sin of X, multiplied by cosine of X, all divided by sin squared of X, which we could go ahead and write that Y. is equal to 2 multiplied by cosine of X divided by sin of X, which is equal to 2 multiplied by cotangent of X when we simplify. And that is our final answer. Hooray, we did it. So therefore, And let's make a note of this. This is equal to Y, which is our first derivative of our function of Y, and we need to also note that for sine of X cannot equal 0. So sin of X cannot equal 0. So that is our final answer 2 multiplied by cotangent of X. Let us also take a moment here briefly to note that we could have taken a different route. We could have noted that why? is equal to 2 multiplied by the natural log of sin of X, then, because if we went that route we could state that Y is equal to 2 multiplied by 1 divided by sine of X multiplied by sine of X, which is equal to 2 multiplied by cosine of X. Divided by sin of x. And as you can see, this would have given us a similar would have given us the same results, so it would have been 2 multiplied by cotangent of X, which leads us to the same answer. So you're also welcome to use the properties of logarithms to simplify and take this approach instead of the approach that I did initially. So whatever methodology makes sense to you could use either or because you will still get the same result. So once again, to quickly recap, in conclusion, the derivative of the function of Y will be. Y is equal to 2 multiplied by the cotangent of X, and that's it. 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(cos² x)

Key Concepts

Derivative

Chain Rule

Logarithmic Differentiation

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