# Thread: How do cos²X and cos X² and cos (X)² all differ derivative wise?!

1. ## How do cos²X and cos X² and cos (X)² all differ derivative wise?!

How are Cos²X and Cos X² and Cos (X)² all differ derivative wise?!

2. ## Re: How do cos²X and cos X² and cos (X)² all differ derivative wise?!

Originally Posted by skinsdomination09
How are Cos²X and Cos X² and Cos (X)² all differ derivative wise?!
Are you clear that $\displaystyle \cos^2(x)~\&~\cos(x^2)$ are two different functions?

Now many computer algebra systems use the notation $\displaystyle \cos(x)^2$ for what traditionally has been written as $\displaystyle \cos^2(x)$.

3. ## Re: How do cos²X and cos X² and cos (X)² all differ derivative wise?!

Originally Posted by Plato
Are you clear that $\displaystyle \cos^2(x)~\&~\cos(x^2)$ are two different functions?

Now many computer algebra systems use the notation $\displaystyle \cos(x)^2$ for what traditionally has been written as $\displaystyle \cos^2(x)$.
Yeah but what confuses me is cos x² vs. cos (x)². Are they different? When parenthesis aren't there I always get mixed up

4. ## Re: How do cos²X and cos X² and cos (X)² all differ derivative wise?!

Originally Posted by skinsdomination09
Yeah but what confuses me is cos x² vs. cos (x)². Are they different? When parenthesis aren't there I always get mixed up
Outside CAS no one uses $\displaystyle \cos(x)^2$. So there is no reason to be confused,

The derivative of $\displaystyle \cos^2(x)$ is $\displaystyle -2\cos(x)\sin(x)~.$

The derivative of $\displaystyle \cos(x^2)$ is $\displaystyle -2x\sin(x^2)~.$

BTW. Because these are functions most of us insist that () be used.
I will mark $\displaystyle \cos~x$ as wrong.

5. ## Re: How do cos²X and cos X² and cos (X)² all differ derivative wise?!

Originally Posted by Plato
Outside CAS no one uses $\displaystyle \cos(x)^2$. So there is no reason to be confused,

The derivative of $\displaystyle \cos^2(x)$ is $\displaystyle -2\cos(x)\sin(x)~.$

The derivative of $\displaystyle \cos(x^2)$ is $\displaystyle -2x\sin(x^2)~.$
Thanks but can you explain the cos²(x) one. It doesn't quite look like either the product or chain so I'm slightly confused. Sorry to be so difficult

6. ## Re: How do cos²X and cos X² and cos (X)² all differ derivative wise?!

Originally Posted by Plato
Outside CAS no one uses $\displaystyle \cos(x)^2$. So there is no reason to be confused,

The derivative of $\displaystyle \cos^2(x)$ is $\displaystyle -2\cos(x)\sin(x)~.$

The derivative of $\displaystyle \cos(x^2)$ is $\displaystyle -2x\sin(x^2)~.$

BTW. Because these are functions most of us insist that () be used.
I will mark $\displaystyle \cos~x$ as wrong.
Sorry but when I look the top one I think chain rule. So i'd derive cos² (the outside) and get -2sin and leave the inside the same so i'd end up with -2sin(X). Why is this wrong?

7. ## Re: How do cos²X and cos X² and cos (X)² all differ derivative wise?!

Originally Posted by skinsdomination09
Thanks but can you explain the cos²(x) one. It doesn't quite look like either the product or chain so I'm slightly confused. Sorry to be so difficult
If $\displaystyle f$ is differentiable function then the derivative $\displaystyle D_x(f^2(x))=2f(x)f^{\prime}(x)$.
It is really the product rule because $\displaystyle f^2(x)=f(x)\cdot f(x)~.$

Thanks!

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