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Problem 29 Hard Difficulty

Find the derivative of the function using the definition of derivative. State the domain of the function and the domain of its derivative.

$G(t)=\frac{1-2 t}{3+t}$

Answer

$\frac{-7}{(3+t)^{2}}$

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Video Transcript

given a function GFT which is equal to one minus duty over three plus T. We need to find G prime of T. Using only the definition of derivative and then you stay at the domain of G and G. Prime of T. Now by derivative definition we have G prime of T. This is equal to the limit As it approaches zero of G. Of T plus H minus G of T. That's all over each. And so we have a limit. It's a job approaches zero of G. Of T plus H. It's the same as one minus T. Time's D plus H over three plus tip. O. Sage- GFT that's 1 -2 T Over three Plus T. And then this all over each. Now simplifying the numerator, combining it as one. We have limit as Asia approaches zero of we have a common denominator of three plus T plus h times three plus T. And then we have three plus t Times 1 -2 times topless H minus. You have 1 -2 T Times three Plus T Plus H. At this time is the reciprocal of eight which is one over H. And then from here we want to simplify the numerator and we get limit as h approaches zero of um we have three plus t minus. We have two times three plus T times T plus h minus. We have three times one minus duty and then minus Tee Times 1 -2 T and then minus Age Times 1- Duty. That's all over three plus T plus H Times three Plus T. And then this multiplied by the reciprocal of a church is one over each. Simplifying further we have limit as H approaches zero of we have three plus t minus yes 60 -6 H minus to t squared minus two th and then minus. We have three plus 60. And then we have minus T plus two T squared and then minus H. Plus to th this all over three plus de plus H Times three Plus T Times one over H. Combining what's left in the numerator We have The limit as h approaches zero of you have negative six H minus H. That's negative seven H over. We have three plus T plus age Times three Plus T. These times one over h. And then from here we can cancel at the age and we get limit as Asia approaches zero of -7 over three plus T plus H times three plus t. Now if we evaluate at age equals zero we have negative seven over three plus T Times three Plus T. Or that's just the same as negative seven over The Square of three Plus T. Therefore this is the derivative of the function before the domain of G. And G. Prime If GFT this is equal to 1- Duty over three plus T. Then we restrict T. Such that three plus T cannot equal zero or that T cannot equal night of three. And so the domain org must be negative Infinity to -3. Union -3 2 Positive infinity, where -3 is not included. And if G prime of T is negative seven over The Square of three busty, then we restrict T. Such that three plus T squared cannot equal zero Or that three plus T cannot equal zero or that T cannot be -3. So it's domain must be the same domain as G. That is from negative infinity to negative three. Union -3 to positive infinity. My negative three is not included.