Derivative of sec(xy) - Stuck on Solving for dy/dx

[LearninDaMath]

Homework Statement

I'm getting stuck on this problem. Not only do I not know how to proceed, I don't understand why I need to put it into that final form.

EDIT: Okay, i get why it needs to be in that final form...obviously because dy/dx needs to be isolated. However, I still don't know how to proceed.

 

[tal444]

Solving this problem involves using the product rule and the chain rule, as well as trigonometric derivatives. Have you learned these yet?

Hint: derivative of sec(x) = sec(x)tan(x)

 

[LearninDaMath]

Solving this problem involves using the product rule and the chain rule, as well as trigonometric derivatives. Have you learned these yet?

Hint: derivative of sec(x) = sec(x)tan(x)

I already derived sec(x). That is not where I'm confused. The part where I'm getting stuck is after the function is already derived and I have to get it into the final form ...after the arrow in the image.

 

[SammyS]

I already derived sec(x). That is not where I'm confused. The part where I'm getting stuck is after the function is already derived and I have to get it into the final form ...after the arrow in the image.

There are several algebra steps involved.

Distribute the sec(xy)∙tan(xy) .

Get all terms with dy/dx on the left hand side. Factor out dy/dx .

...

 

[tal444]

I already derived sec(x). That is not where I'm confused. The part where I'm getting stuck is after the function is already derived and I have to get it into the final form ...after the arrow in the image.

What I'm trying to say is that the way you derived it is probably making it more difficult. Use sec(xy)tan(xy) and use the chain rule for the (xy). Then follow what SammyS said.

 

[Mark44]

LearninDaMath,
You are making the same mistake you made on a thread of few days ago. The first line in your post says
"y = sec(xy)"
This makes no sense because the right side also involves y.

Your work should look something like this:

d/dx(sec(xy)) = sec(xy)*tan(xy) *d/dx(xy)
= ...

For the derivative at the end of the first equation above, you need to use
1) the product rule, followed by
2) the chain rule

Tip: make life a little easier on yourself by using the fact that d/dx(sec(x)) = sec(x)*tan(x). You are making the problem more difficult than it needs to be by not using this formula.

 

[LearninDaMath]

tal444, thanks for your help. Sorry that my question may not have been specific enough. Sometimes it feels more efficient for me to scan my work directly to the computer instead of trying to figure out the latex symbols. However, I am getting better at it though.

SammyS, thanks. I got it now. Appreciate your feedback as always.

 

[Mark44]

tal444, thanks for your help. Sorry that my question may not have been specific enough. Sometimes it feels more efficient for me to scan my work directly to the computer instead of trying to figure out the latex symbols. However, I am getting better at it though.

But it's not more efficient for us, since we can't insert a comment in at exactly the right spot.

SammyS, thanks. I got it now. Appreciate your feedback as always.

 

[tal444]

No problem, glad I could help.

 

[LearninDaMath]

LearninDaMath,
You are making the same mistake you made on a thread of few days ago. The first line in your post says
"y = sec(xy)"
This makes no sense because the right side also involves y.

Your work should look something like this:

d/dx(sec(xy)) = sec(xy)*tan(xy) *d/dx(xy)
= ...

For the derivative at the end of the first equation above, you need to use
1) the product rule, followed by
2) the chain rule

Tip: make life a little easier on yourself by using the fact that d/dx(sec(x)) = sec(x)*tan(x). You are making the problem more difficult than it needs to be by not using this formula.

Mark, I was typing my comment during the time you posted yours. Just saw that your post now, thanks for providing information on this. The question of my post comes directly from the issued textbook. The question is like this:

Find dy/dx by implicit differentiation.

sec(xy) = y

and that is exactly as its written.

I recall your feedback from that previous posted saying that I can not refer to sec(xy) = y as: f(x) = sec(xy)

and this is because sec(xy) is not a function. However, it is still an equation. And implicit differentiation can be applied to both sides of an equation, even if that equation involves the same variable on both sides.

I believe this was what I took away from the recent help you provided on that previous thread.

 

[LearninDaMath]

But it's not more efficient for us, since we can't insert a comment in at exactly the right spot.

Oh, I didn't see it from the view of those who are providing the instruction/assistance, good point. I'll start transitioning over to latex format.

 

[Mark44]

Mark, I was typing my comment during the time you posted yours. Just saw that your post now, thanks for providing information on this. The question of my post comes directly from the issued textbook. The question is like this:

Find dy/dx by implicit differentiation.

sec(xy) = y

and that is exactly as its written.

And this is why we ask that you use the template, with the first part being the problem statement. In essence, the problem is asking you to implicitly differentiate the equation y = sec(xy). The equation is not defining y as a function of x and itself, which is what I thought you were trying to do.

I recall your feedback from that previous posted saying that I can not refer to sec(xy) = y as: f(x) = sec(xy)

and this is because sec(xy) is not a function.

But it is a function - a function of two variables.

However, it is still an equation. And implicit differentiation can be applied to both sides of an equation, even if that equation involves the same variable on both sides.

I believe this was what I took away from the recent help you provided on that previous thread.

Yes, good.

 

[Mark44]

Oh, I didn't see it from the view of those who are providing the instruction/assistance, good point. I'll start transitioning over to latex format.

Thank you. I appreciate it, and I think others will as well.

LaTeX isn't very hard, and if all you need to do are exponents, you can do them without LaTeX. Just click the Go Advanced button below the input area, which opens an advanced menu along the top of the input area. The X² can be used for exponents, and there are a bunch of useful symbols along the right side, including ∫, Ʃ, ∞, √, and several others.