Calculate the derivative of the following functions. In some c...

Question

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 = (cos x) In cos²x

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. Calculate the derivative of the function y equals sin of x multiplied by the natural log of squad of X. Awesome. So it appears for this particular problem, what we're ultimately trying to do, what we're trying to solve for is we're trying to figure out what the derivative of this function of Y is. So in order to do that, first off, we need to recall and use the following rule which states that log. Base A of B to the power of X is equal to X multiplied by log base A. Of B So when we recall and apply this rule, we can simplify our function of Y, and we can use it to help us simplify the overall differentiation process that we're trying to perform. So when we do that, when we apply this log rule, we will find that Y is equal to parenthesis of X multiplied by 2 multiplied by the natural log of sign of X. Fantastic. So now, and that was our first step. Our first step is what we needed to recall what the log rule was and then apply it to our function. Now our second step that we need to take is we now need to let the letter F equal sign of X, and we need to let G equal 2 multiplied by the natural log of sin of X. So the function Y, as we should recall in note, focusing on our particular function for Y, the function of Y will be a product of two functions. Thus we can recall and apply the product rule. So we can therefore state that Y and this prime symbol, so this like power to like this asterisk, this means that it's the derivative, the first derivative. So, Y. Equals parenthesis F multiplied by G. So it's F multiplied by Gha to the power of prime, so prime. equals F multiplied by G plus F multiplied by G. So, when we apply this to our function, we can say that Y is equal to sine of X multiplied by. 2 multiplied by the natural log of sin of x, plus sin of x multiplied by 2 multiplied by the natural log prime of sin of X. Fantastic. So now our third step that we need to take is we need to evaluate the derivatives, and as we should recall and note that for sine x or sine of X, we can use the table of basic derivatives, so meaning, as we should recall, we could state that sign, so we could say that sine prime of X, so sine prime. of X is equal to cosine of X, and for the natural logarithm, we need to recall and note that the natural log prime of X is equal to 1 divided by X. So in this case, as we should recall a note, We will have a composite function, so we'll need to recall and apply the chain rule in order to perform this derivative. So, as we should recall, we apply and use the chain rule, we'll be able to write that D by DX. Of the natural log of sin of X is equal to 1 divided by sin of X, multiplied by D by DX of sine of X, which is equal to 1 divided by sin of X, multiplied by cosine of X, which is all ultimately equal to cosine of X. Divided by sin of X. Fantastic. So now our 4th and final step is we need to substitute our results from step 3 into the expression that we got in step 2. So when we do that, we will find that Y is equal to 2 multiplied by cosine of X multiplied by the natural log of sine. Of X plus 2 multiplied by sin of X multiplied by cosine of X divided by sine of X. Fantastic. And when we do a little bit of simplifying, we will find that Y is equal to 2 multiplied by cosine. Of X multiplied by the natural log of sin of X plus 2 multiplied by cosine of X, but we can simplify even we can simplify even more, and we could simplify it down to state that Y is equal to, or rather we can simplify one last time and we can state that Y is equal to 2 multiplied by cosine. Of X multiplied by the natural log of sine of X. Plus one. And without a doubt, this will be our final answer for the derivative of the function for Y. Hooray, we did it. So therefore, without a doubt, our final answer for our derivative, once again for this function of Y will be Y is equal to 2. Multiplied by cosine of X, multiplied by the natural log of sin of x, plus 1. And that's it. We've officially solved for this problem. Thank you so much for watching. Hopefully that helped, and I can't wait to see you in the next video. Bye.

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 = (cos x) In cos²x

Key Concepts

Derivative

Logarithmic Properties

Chain Rule

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