FIND THE DERIVATIVE OF THE FUNCTION

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So in this problem we were given the function GFX equals integral from 0 to X Square root of two plus T cubed DT. And we have to find the derivative of G effects. Now, if we use the fundamental theorem of calculus then we can immediately see that derivative of G x g prime X would be simply Square root of two plus x cubed. But if we don't use the fundamental theorem of calculus, we can show it from just the definition of the veterans as well. So recall that the derivative of G at X is given by the limit limit of H tends to zero G F X plus H minus G f x over H. Now no dot G F X plus H minus G f X over H Is simply 1/8 and G f x plus H would be integral of 02 X plus h squared of two plus t cubed DT and we have minus G fx and vfx is simply integral of 02 X two plus t cubed DT. So and this this expression we can see that we can write it as one of our age integral from X to X plus h square root of two plus D Q two DD. So here the integral is from zero to explicit and here it is from zero to X. So if we subtract this from this and then then we are left with the integral from X to X plus H squared of two plus two cubed. And now here No. of two. Plastic Cube is an increasing function. And here four square root of two plus two cubed. To make sense we must have that T cubed plus two is greater than equal to zero. That is T. Is greater than equal to cube root of two. Ah greater than equal to minus cube root of two. So that's implicitly kept in mind. So here we have one of our agent to grow from extra explicit square root of two plus two Q. Two D. T. And sends to the square root of two plus two cubed is an increasing function. We have that here in this interval T. Is greater than equal to X. And less than equal to X plus H. So we have squared of two plus X cubed times one of 1/8 integral from X to X plus A C. D. T. Is less than equal to 1/8 integral from X to x plus H squared of two plastic cube dd. And then we use that the in this interval is less than equal to X plus eight. So two plus square root of two plus two cubed is less than equal to square root of two plus X plus H hold queue So this is less than equal to square root of two plus explicit. Hold cubed times. We have the one over age times the integral from X to X plus H. D. T. Now this weekend right at Square root of two plus x cubed times X plus H minus X over age is less than equal to one over age integral from X to X plus H squared of two plus T cubed DT and that is less than equal to two plus X plus H cubed times X plus H minus X over age that is two plus X squared of two plus X cubed. So here explicit H minus X is age and that divided by H is one. So this becomes one. So we have square root of two plus X cubed is less than equal to one over age. Times integral from X to X plus H two plus t cubed squared of two plus T cubed DT And that is less than equal to two plus X plus H hold cute one here again as well, X plus H minus X over H as one. And so from here No data as h tends to zero this term does not have any age so it's constant. So the limit as H tends to zero squared of two plus executed is simply itself and the right and bound we have the limit as H tends to zero of this is also two plus execute Here, H. tends to zero. So both the upper and lower bounds tend to the same limits. So by the squeeze lemma this this limit limit of H tends to zero of this middle thing without also equals square root of two plus x cubed. That is we can write G. Prime Mexico was limit of H tends to zero G. X plus H minus G. X over H is the limit of H tends to 0 1/8 times integral from extra X plus H squared of two plus T cubed DT. And that, like we said already, by the squeeze lemma is equal to Square root of two plus x cubed and that's the derivative of G. And that's it for this problem.

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