Solve Derivative of sin^2(pie*Z) with respect to Z

[innightmare]

Homework Statement

What is the derivative of sin^2(pie*Z) with respect to Z

Homework Equations

The Attempt at a Solution

I think the answer is pie/2*sin(pie*Z)

Is this correct? I keep getting confused with whether or not I should involve the 2 or not of should just leaving it along and just focus on my angle, Z

 

[dr3vil704]

sin^2(pie*Z) is sin^{2}(\piz)?
If it is, Then can't you use the chain rule? which is power rule on the whole out side of the ( ) and then it by multiply by the derivatives of the inside of the ( ). because sin^{2}(\piz) is just the same as (sin(\piz))^{2}

 

[rocomath]

(\sin{x})^2 = \sin^{2}x

 

[dr3vil704]

yeah, so Use the chain rule. Look at it as (sinx)^2 because is easier if you look at it like this.

 

[innightmare]

no pie is inside the paranthesis next to Z. I was thinking of doing the chain rule but wasnt sure. but its sin -square and the angle is pie next to Z

Is the answer sin over two *pie

 

[rocomath]

\frac{d}{dz}\sin^{2}{(\pi Z)}

 

[rocomath]

no pie is inside the paranthesis next to Z. I was thinking of doing the chain rule but wasnt sure. but its sin -square and the angle is pie next to Z

Is the answer sin over two *pie

no, chain rule?

 

[dr3vil704]

hm..I believe the derivative of sin is cos. so the answer should be 2cos(pi*Z) * (Pi) or 2\picos(\piZ).

the first part is power rule and derivative of the Sin which is Cos. for the inside of the ( ), since it's product of a variable and a constant, we know that Pi is a number hence it a constant. So you take the dervitive of pi*Z and use the product rule:
it will be
Pi*1 + Z*(0)=Pi.

 

[jacobrhcp]

no.
chain rule.

>_> just do the derivative of x^2 with respect to x [x=sin(pi z)] (the deravative is 2x). After that multiply by the derivative of x with repect to z. d/dz (sin(pi z)) = pi cos(pi z).

and this is: 2 pi sin(pi z)cos(pi z) and I think that's also equal to pi sin(2pi z) by trigonomtric identities, but not sure about that.

 

[jacobrhcp]

never mind this post

 

[Saladsamurai]

hm..I believe the derivative of sin is cos. so the answer should be 2cos(pi*Z) * (Pi) or 2\picos(\piZ).

the first part is power rule and derivative of the Sin which is Cos. for the inside of the ( ), since it's product of a variable and a constant, we know that Pi is a number hence it a constant. So you take the dervitive of pi*Z and use the product rule:
it will be
Pi*1 + Z*(0)=Pi.

NO product rule! \pi is NOT a variable, it is a constant just like 2 or \frac{3}{4} or any other NUMBER.


The chain rule qualitatively says: Take the derivative of the 'outside function' with the inside function as its argument and multiply it times the derivative of the 'inside function'.

In this case there are 2 outside functions. Start with the squared function and work your way inwards.

Casey