#### David98678

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- Thread starter David98678
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I do not understand why you would write cosx

Here is the rule you must use. Suppose u is anything. Then \(\displaystyle \frac{d}{dx}\)(u)

Please note that in this example, u is simply cos(x) and n= -2.

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The notation \(\displaystyle \cos\) stands for the

That is use parentheses, \(\displaystyle \large\cos(x)\). Even though historically the notation \(\displaystyle \cos x\) was used, in practice we have move away from that.

If we have a function \(\displaystyle f(x)\) you surely know that there is a difference in \(\displaystyle f(x^2)~\&~f(x)^2\).

So that \(\displaystyle \cos^2(x)=\cos(x)^2\)

If you still have confusion on this notation please post your questions.

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So, the problem is to differentiate \(\displaystyle \dfrac{1}{\cos^2 x}\)?

I would write it as \(\displaystyle (\cos x)^{-2}\) and use the chain rule. That is, take \(\displaystyle u = \cos x\), differentiate \(\displaystyle u^{-2}\), and multiply that by the derivative of \(\displaystyle \cos x\).

What do you get?

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Okay, I give up! I know what \(\displaystyle \cos^2(x)\) means and what \(\displaystyle cos(x^2)\) means. But what in the world is \(\displaystyle cos(x)^2\) if it is not a sloppy way of writing one of the others?The notation \(\displaystyle \cos\) stands for thetherefore we should usecosine function.function notation

That is use parentheses, \(\displaystyle \large\cos(x)\). Even though historically the notation \(\displaystyle \cos x\) was used, in practice we have move away from that.

If we have a function \(\displaystyle f(x)\) you surely know that there is a difference in \(\displaystyle f(x^2)~\&~f(x)^2\).

So that \(\displaystyle \cos^2(x)=\cos(x)^2\)BUT\(\displaystyle \cos^2(x)\ne\cos(x^2)\ne\cos(x)^2\).

If you still have confusion on this notation please post your questions.

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Come on Prof Halls, \(\displaystyle \cos(x^2)\) is the cosine of the square of x , in the same way any function \(\displaystyle f(x^2)\) means \(\displaystyle f\) evaluated for \(\displaystyle x^2\).Okay, I give up! I know what \(\displaystyle \cos^2(x)\) means and what \(\displaystyle cos(x^2)\) means. But what in the world is \(\displaystyle cos(x)^2\) if it is not a sloppy way of writing one of the others?

That is a good reason for using \(\displaystyle \cos(x)\) in stead of \(\displaystyle \cos~x\).

Many calculators, basic programming,and computer algebras require the notation \(\displaystyle \cos(x)^2\) (actually any fuction) to return the square of the function, i.e. the routine does not recognize \(\displaystyle f^2(x)\) for \(\displaystyle f(x)^2\).

BTW: The post you quoted carefully said \(\displaystyle \cos^2(x)=\cos(x)^2\)

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… what in the world is \(\displaystyle cos(x)^2\) …

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So we can't calculate sin(cos(x))???cos(x)is a number

cos(x)is the number^{2}cos(x)being squared

The former is poor form if it is intended to be equivalent for the square of the cosine of x, because it hascos(x)is the number^{2}cos(x)being squared

Extra grouping symbols should be used for it instead, such as:

\(\displaystyle [cos(x)]^2\)

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It's not ambiguous, for folks who understand function notation. Is your position that it shouldn't be used because you think a majority of people do not understand function notation? :cool:… it hasambiguousstyle. It should not be used …

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I don't understand your statement (followed by question marks). Do you have a value for x?So we can't calculate sin(cos(x))???

No, I claim it is not consistent function notation for the exponentiationIt's not ambiguous, for folks who understand function notation. Is your position that it shouldn't be used because you think a majority of people do not understand function notation? :cool:

of trig functions to begin with.

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I agree that \(\displaystyle \cos(x)^2\) isThe former is poor form if it is intended to be equivalent for the square of the cosine of x, because it hasambiguousstyle. It should not be used.

Extra grouping symbols should be used for it instead, such as:

\(\displaystyle [cos(x)]^2\)

In a context where one

But in trig and log problems in textbooks, it is still common to use the old, pre-function notation where \(\displaystyle \cos x\) or \(\displaystyle \log x\) is acceptable. And in that context, consider a slightly worse case: \(\displaystyle \cos(x+1)^2\). Here, the parentheses might be present to mark the argument, \(\displaystyle x+1\); or they might be there to distinguish the base of the power, the whole thing being the argument.

In an ideal world, a little thought may make it clear that the parentheses are intended to hold the argument; but expecting everyone to think carefully is "blaming the victim". Consideration for the reader demands that we remove stumbling blocks in case they are not accustomed to seeing things the same way we do. That's how good communication works: the transmitter plans the content to prevent errors on the part of the receiver, rather than putting the entire burden on the receiver.