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

In summary, the conversation was about finding a derivative using two different methods. The function given was y=10*sinpi(.01x-2.00t). One method involved multiplying pi into the parentheses while the other method treated pi as a multiplier. The output from both methods were the same, but one was simpler than the other.
  • #1
paul11273
156
0
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.
 
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  • #2
You can certainly multiply it out - it's just the distributive property of numbers.
 
  • #3
However, it is simpler just to treat [itex]\pi[/itex] as a multiplier:

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

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

Finally, the derivative of [itex]sin(\pi(0.01x- 2.00t))[/itex] is, of course, just 10 times that: [itex]-20.00\pi cos(\pi(0.01x-2.00t)[/itex].
 
  • #4
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.
 

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

What is a "weird derivative conflict"?

A "weird derivative conflict" occurs when there are multiple possible derivatives for a function at a specific point. This can happen when the function is not continuous or differentiable at that point, or when there are multiple branches or discontinuities in the function.

How do you resolve a "weird derivative conflict"?

To resolve a "weird derivative conflict", you must first determine the cause of the conflict. This may involve graphing the function, identifying any discontinuities or points of non-differentiability, and understanding the behavior of the function near the point in question. Once the cause is identified, you can use mathematical techniques such as the limit definition of a derivative or the quotient rule to find the correct derivative.

What are some common sources of "weird derivative conflicts"?

Common sources of "weird derivative conflicts" include functions with sharp corners, cusps, or vertical tangents, as well as functions with piecewise definitions or absolute value terms. Additionally, functions with oscillating behavior or points of discontinuity can also lead to "weird derivative conflicts".

Why is it important to resolve "weird derivative conflicts"?

Resolving "weird derivative conflicts" is important because the derivative is a fundamental concept in calculus and is used in many applications to find the rate of change or slope of a function. Having incorrect or undefined derivatives can lead to incorrect results and limit the usefulness of the derivative in understanding the behavior of a function.

What strategies can you use to avoid "weird derivative conflicts"?

To avoid "weird derivative conflicts", it is important to carefully analyze the behavior of a function and identify any potential points of non-differentiability or discontinuities. Additionally, simplifying or rewriting functions with absolute value or piecewise definitions can help avoid potential conflicts. Finally, using mathematical techniques such as the limit definition of a derivative can also help avoid "weird derivative conflicts" by providing a more precise and accurate calculation of the derivative.

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