Which step is wrong in deriving the derivative of sec(x)?

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onesun0000

Junior Member
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I asked a friend to determine which step, if there is, has an error in deriving the derivative of sec (x). I only know how to use the quotient rule to derive it that's why I got confused. Here's the actual question:

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Here's my friend's answer, but I still don't get it.
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It's quite difficult to respond to this post. Our aim is to increase YOUR understanding, that's why we ask you to show YOUR work so that we can see where you are stuck. However you have shown a friend's work and stated that you don't understand either the original question or your friend's answer.

So, to start off, please tell us which "step" (step 1, step 2, etc) of the original question you don't understand and try to tell us why you don't understand it so that we can help YOU out.
 
There is no error in what you have done. The problem specifically says "where x is in the domain of sec(x)" which means cos(x) is not 0 so dividing by cos(x) is valid.

As for your friends argument about applying the product rule to \(\displaystyle |x^2|\), the product rule specifically requires that f and g separately be differentiable but |x| is not differentiable at x= 0.
 
I second Cubist’s reply, but have some additional comments.

You seem to believe that the derivation is in error. Why do you think so? What do you think the derivative of sec(x) is?

What is the course in which this problem is posed?

It is of course true that the secant is not everywhere defined and that therefore its derivative is not everywhere defined. But that is true whether you use the quotient rule, the power rule, or the product rule. So your friend’s answer is completely irrelevant.
 
Cubist said:
It's quite difficult to respond to this post. Our aim is to increase YOUR understanding, that's why we ask you to show YOUR work so that we can see where you are stuck. However you have shown a friend's work and stated that you don't understand either the original question or your friend's answer.

So, to start off, please tell us which "step" (step 1, step 2, etc) of the original question you don't understand and try to tell us why you don't understand it so that we can help YOU out.
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First of all, maybe this will help, I know how to derive the derivative of sec(x) using quotient rule. In the steps, I understand step 1 because thats like multiplying by reciprocal that's why it's equal to 1. In step two, since [MATH]\sec(x) \cos(x)=1[/MATH], its derivative is zero because the drivative of a constant is zero. In step 3, that's where the product rule is used, which is used correctly. In step 4, I quite understand it, in a sense that I think quotient rule is used there. That's why I don't understand if there is an error or not. It seems no error at all. But my gut says there is. I am not sure.
 
JeffM said:
I second Cubist’s reply, but have some additional comments.

You seem to believe that the derivation is in error. Why do you think so? What do you think the derivative of sec(x) is?

What is the course in which this problem is posed?

It is of course true that the secant is not everywhere defined and that therefore its derivative is not everywhere defined. But that is true whether you use the quotient rule, the power rule, or the product rule. So your friend’s answer is completely irrelevant.
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My friend also said this.
1603494187532.png
 
The proof by your friend is fine. I felt it was an unusual way of doing it but that's my problem.

I (all of us I am sure) would like to see your proof that you speak of using the quotient rule. Can you please post it.

Personally I would never use the quotient rule if the numerator or denominator is a constant. sec(x) = 1/cos(x) = (cos(x))-1. I would use the power rule from here to compute the derivative of sec(x). Give it a try and post back with your work.
 
onesun0000 said:
In step 4, I quite understand it, in a sense that I think quotient rule is used there.
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Step3) [MATH]\dfrac{d}{dx}sec(x)*cos(x)-sec(x)*sin(x) = 0[/MATH] so [MATH]\dfrac{d}{dx}sec(x)*cos(x)=sec(x)*sin(x)[/MATH] then [MATH]\dfrac{d}{dx}sec(x) \dfrac{sec(x)*sin(x)}{cos(x)} = sec(x)*tan(x)[/MATH].

There was no calculus done above at all. Just pure algebra solving for[MATH]\dfrac{d}{dx}sec(x)[/MATH]
 
Jomo said:
The proof by your friend is fine. I felt it was an unusual way of doing it but that's my problem.

I (all of us I am sure) would like to see your proof that you speak of using the quotient rule. Can you please post it.

Personally I would never use the quotient rule if the numerator or denominator is a constant. sec(x) = 1/cos(x) = (cos(x))-1. I would use the power rule from here to compute the derivative of sec(x). Give it a try and post back with your work.
Click to expand...
That's how I use quotient rule to derive the derivative of [MATH]\sec(x)[/MATH]. So is there an error on the steps?
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onesun0000 said:
oh

so the error is on step 3?
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A fine proof has no errors!
In an earlier post you said that you did not understand how to go from step 3 to step 4 so I filled in some of the missing steps. There were no errors in the original proof which you posted.
 
Jomo said:
A fine proof has no errors!
In an earlier post you said that you did not understand how to go from step 3 to step 4 so I filled in some of the missing steps. There were no errors in the original proof which you posted.
Click to expand...
Jomo

I think the friend said the proof was wrong.
 
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