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Find the derivative of the function using the definition of a derivative. State the domain of the function and the domain of its derivative.$f(t)=5 t-9 t^{2}$

domain of the function $(-\infty, \infty)$domain of the derivative $(-\infty, \infty)$

Calculus 1 / AB

Chapter 3

Derivatives

Section 2

The Derivative as a Function

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already. So again, we're going to use the definition of the derivative, which is tthe e f Prime of X is equal to the limits as h approaches zero uh, the function X plus H minus f of X over h and in this instance, our function R f and they gave it in terms of tea is gonna be five t minus T squid. How does this going? Minus 90 square 90 square. Okay, so that means that f of tea plus age will be equal to five times tea plus H minus nine T plus age square. Okay. And that's actually gonna be equivalent to, So we'll distribute the 55 t plus five h have minus nine t squared, minus 18 th minus nine h squared. So we can plug both of these into our definition. And that would be the limit as h purchase zero five T plus five age minus nine. T squared minus 18. Th minus nine h squared all over. H, okay. And so this is gonna be equal to the limit as age approaches. Zero. Ah, negative. Nine h squared. Plus five minus 18. He times age all over h. And what I forgot to do the top. Of course. Subtract my f of X rights. I wrote my f of X plus h but we should also have a minus five T and an A plus mind t squared. And so that would have cancelled out five teas and the 90 squares, OK? And so we can't see it take in a child. Right? So this age can come out and that one and so this is gonna be than equal to the limit as H approaches zero of negative nine H plus five minus 18 t. Okay, well, h goes to zero. This is simply going to be equal to your neighbor. 19 become zero, so it's gonna be able to five minus 18 t. Okay, so this is our derivative f prime. And now we want to know what the domain is of each of these. So our function original function five T minus t squared. This is defined everywhere. So the domain is negative. Fate to infinity and as well as our derivatives. There's nothing you can put in here that would make undefined. So again, the domain is negative. Infinity to infinity. All real numbers

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