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

**skinsdomination09** · October 1st 2012, 01:43 PM

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

**Plato** · October 1st 2012, 01:49 PM

Originally Posted by skinsdomination09

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

Are you clear that $\cos^2(x)~\&~\cos(x^2)$ are two different functions?

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

**skinsdomination09** · October 1st 2012, 01:52 PM

Originally Posted by Plato

Are you clear that $\cos^2(x)~\&~\cos(x^2)$ are two different functions?

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

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

**Plato** · October 1st 2012, 02:04 PM

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 $\cos(x)^2$ . So there is no reason to be confused,

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

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

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

**skinsdomination09** · October 1st 2012, 02:06 PM

Originally Posted by Plato

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

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

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

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

**skinsdomination09** · October 1st 2012, 02:11 PM

Originally Posted by Plato

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

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

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

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

**Plato** · October 1st 2012, 02:13 PM

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 $f$ is differentiable function then the derivative $D_x(f^2(x))=2f(x)f^{\prime}(x)$ .
It is really the product rule because $f^2(x)=f(x)\cdot f(x)~.$

**skinsdomination09** · October 1st 2012, 02:21 PM

Thanks!

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