Is There a Simpler Way to Handle the Pi in Finding a Derivative?

[paul11273]

I am having a conflict with two different ways of finding a derivative.
Here is the function:
y=10*sinpi(.01x-2.00t)

Yes, that pi is after sin, but not in the paranthesis. This is how the prof gave it to us. This may be my problem, how I am treating the pi. I figure it was factored out of the parenthesis. So, to find the partial derivative WRT t by hand I do this:

y=10*sin(.01pi*x - 2.00pi*t) I multiplied the pi into the ()
dy/dt = -2.00pi*10*cos(.01pi*x - 2.00pi*t) used the chain rule
dy/dt = -20pi*cos(.01pi*x - 2.00pi*t) final result

That is my result. I check this in Matlab by entering the following:

>> syms x t
>> diff(10*sin(pi*.01*x-pi*2*t),t)
ans =
-20*cos(1/100*pi*x-2*pi*t)*pi

So with that I am happy. Now the tricky question.
If I enter this same thing to my TI-89, I get:
-62.8319*cos(2pi*t - .031416*x)

Now...it just hit me that you can transpose the items inside the paranthesis of cosine, and it is the same result. Ok, duh. I don't want to delete everything I just typed. My next question...

Am I treating the pi correctly to begin with? Is it correct to multiply it into the () like that? If not, what should I do with it? Is there an easier way?
Thanks.

 

[Tide]

You can certainly multiply it out - it's just the distributive property of numbers.

 

[HallsofIvy]

However, it is simpler just to treat $\pi$ as a multiplier:

The derivative of sin x is cos x so the derivative of $sin(\pi x)$ is $cos(\pi x)$ times the derivative of $\pi x$ (that's the "chain" rule) which is just $\pi$: the derivative of $sin(\pi x)$ is $\pi cos(\pi x)$.

The partial derivative of $sin(\pi(0.01x- 2.00t))$, with respect to t, is just that times the derivative of 0.01x- 2.00t with respect to t: $-2.00\pi cos(\pi (0.01x- 2.00t)$.

Finally, the derivative of $sin(\pi(0.01x- 2.00t))$ is, of course, just 10 times that: $-20.00\pi cos(\pi(0.01x-2.00t)$.

 

[paul11273]

Thanks

Thank you both for responding.
I see what Halls has done, and yes, I think that is simpler. It just didn't occur to me to put pi into the () one time, instead of multiplying to each term.
Thanks.