# What is the derivative of the function `T(t)`? `T(t)=(4t/t^2+1)+98.6`

txmedteach | High School Teacher | (Level 3) Associate Educator

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We're looking for the derivative of the above function. We can actually easily simplify the function. After acknowledging this hole in the domain, we can go ahead and cancel the `t`'s in the function to get the following:

`T(t) = 4/t + 1 + 98.6`

Now, we can combine the constant terms:

`T(t) = 4/t + 99.6 = 4t^(-1) + 99.6`

We can now differentiate easily. Recall, the derivative of the above function will be like any polynomial: bring down the -1, and the resulting power will be `t^-2`. The derivative of the constant term will, of course, go to zero.

`(dT)/(dt) = -4/(t^2)`

There you have it! There really isn't much more from here.

Normally, you need to worry about domain issues when you cancel out variables like we did in the first step; however, the domain in our case did not change. You need to watch out for when the original function does not contain a certain number in the domain before you cancel out values. For example, if you're given the function:

`f(x) = ((x-6)(x-5))/(x-6)`

You can cancel the `(x-6)` term, but you need to remember that `x = 6` is not in your domain. So just watch out for things like that!

I hope that helps!

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