In Exercises 25–30, use logarithmic differentiation to find th...

Question

In Exercises 25–30, use logarithmic differentiation to find the derivative of y with respect to the appropriate variable.

25. y = 2(x² + 1)/√(cos 2x)

Accepted Answer

Hello there. Today we are 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. Use logarithmic differentiation to compute Dy by DX4 Y is equal to 4 multiplied by the parenthesis of X2 minus 3, all divided by the square root of sin of 2 X. OK. So it appears for this particular pro we're asked to use logarithmic differentiation to compute Dy by DX for our for our equation of Y that we are given by the problem itself. So now that we know what we're ultimately trying to solve for and what method we're asked to use to solve this problem is we're using logarithic differentiation. We're gonna be requiring our prior knowledge of logs and how to differentiate logs. Or log-based expression. So using our prior knowledge, our first step that we need to take. we need to recall and take the natural logarithm of both sides of our equation for Y. So we need to take the natural log of Y is equal to the natural log of Once again, the natural log of 4 multiplied by parentheses X to the power of 2 minus 3, all divided by the square root of sign of 2 X. Fantastic. So our second step now is we need to recall how to use log properties, so we need to recall our prior knowledge about log properties, and when we do, we need to be able to recall that we could write that the natural log of Y is equal to the natural log of 4 plus the natural log of parenthesis. X to the power of 2 minus 3, minus 1/2, 1 divided by 2 multiplied by the natural log of parenthesis, sign of 2 X. fantastic. So, our third step, this is step number 3. We now need to differentiate both sides with respect to X, so we need to take 1 divided by Y, D Y by DX, so 1 divided by Y multiplied by Dy by DX, which is equal to 0 + 2 X. Divided by X2 minus 3, and then we need to subtract 1 divided by 2 or 12. Multiplied by 1 divided by sin of 2 X, multiplied by cosine of 2 X multiplied by once again it's gonna be cosine of 2 X multiplied by 2. Fantastic. Our fourth step now is we need to simplify our derivatives, so we need to take 1 divided by Y, multiplied by Dy by DX is equal to 2 x divided by X2 minus 3 minus cosine of 2 x divided by sin of 2 X. Fantastic. So, with that said, Our next step, so this will be step number 5 now. So our 5th step we need to recognize that cosine of 2 X divided by sin of 2 X will be equal to cotangent of 2 X. So, that will mean that 1 divided by Y multiplied by Dy by DX will be equal to 2 x divided by X2 minus 3 minus cotangent of 2 X. Fantastic. And our 6th steps, this is step number 6, is we need to multiply both sides by Y. and when we do, we will find that DY by DX will be equal to Y multiplied by the parenthesis of 2 X divided by X2 minus 3 minus cotangent of 2 X fantastic. And our 7th step, which is our last step, is we need to substitute, so I'll make a star here. We need to substitute Y is equal to 4 multiplied by parenthesis of X2 minus 3 divided by the square root of sin of 2 X. Into our expressions, once again, we're substituting in our value of Y, and this is the value of Y that is given to us by the problem itself. So, when we substitute in our value of Y, we will be able to write that Dy by DX is equal to 4 multiplied by parenthesis of X to the power of 2 minus 3 divided by the square root of, of 2 X multiplied by parenthesis, 2 x divided by X2 minus 3. Minus cotangent of 2 X fantastic. And this will be equal to drum roll or multiplied by X2 minus 3. All divided by the square root of sin of 2 X, all multiplied by parenthesis of 2 X minus X to the power of 2 multiplied by cotangent of 2 X plus 3 multiplied by cotangent of 2 X. All divided by. Next to the power 2 minus 3. Fantastic. And this will be simply equal to 4 divided by the square root of sin of 2 X multiplied by the parenthesis of, which you can use a little dot to indicate a multiplication is occurring, but you could just write it without the dot. So We're going to find out when we simplify, we'll find that it's equal to 4 divided by the square root of sin of 2 x multiplied by the parenthesis of 2 X minus X to the power 2 multiplied by cotangent of 2 X plus 3 multiplied by cotangent of 2x, and this is our final answer. And we did it. So to quickly recap what we've learned, so going back to look at the prom itself, we can say without a doubt that once again, our final answer is as follows. So, Squeeze our answer in here so you can see. So our final answer once again, without a doubt will be Dy by DX is equal to 4 divided by the square root of sin of 2 X multiplied by parenthesis of 2X minus x 2 multiplied by cotangent of 2 X plus 3 multiplied by cotangent of 2 X. And that's it. That is our final answer. Thank you so much for watching. Hopefully that helped and I can't wait to see you in the next video. Bye.

In Exercises 25–30, use logarithmic differentiation to find the derivative of y with respect to the appropriate variable.25. y = 2(x² + 1)/√(cos 2x)

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In Exercises 25–30, use logarithmic differentiation to find the derivative of y with respect to the appropriate variable.
25. y = 2(x² + 1)/√(cos 2x)