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Problem 54 Hard Difficulty

Use the definition of a derivative to find $ f'(x) $ and $ f''(x) $. Then graph $ f $, $ f' $, and $ f'' $ on a common screen and check to see if your answers are reasonable.

$ f(x) = x^3 - 3x $

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Video Transcript

This is a problem. Number fifty four of the Stuart Calculus eighth Edition Section two point eight. Use the definition of a derivative to find at prime of X and F double primary Kes, then Graff. Time enough. Double prime on a common screen and check to see if your answers are reasonable and the function F it's given us at of X equal to execute minus X minus three X name. So let's start with and paramedics by using the derivative ah, definition of a derivative. The prime of X is given as the limit as each of purchase. Zero of the function hearing Next Hugo Industry X Evaluated Expo Sage. So the text plus age quantity cubed minus three times his age. And then we're going to subtract the function execute ministry and this on the right of right teach. Okay. Our next step is to expand the numerator. Separate all the terms X plus h Cubed is binomial cubed will give us X cubed plus three expert H plus three x h squared not plus age cubed. Then we distribute the negative three in the next term, making three expense three each and then we should make it appear negative, X cubed Class three X and they're all divided by each. Hey, we have a next cute how you hear minus and execute high here. Ah, a positive three x here and a negative directs there that were canceled and then each remaining term has it one age values. So the agent the denominator counsels with one h and each of the remaining terms. And that leaves us with for Lim as each approaches zero out of the function three x squared less three x h pompous h squared minus three. And if we take a look here each term, that has an a jewel that being negligible since ages approaching zero. So three x h and H squared will go to zero and the resulting terms are are derivative three X squared minus three. Now, for the second derivative, we do the same procedure except we are using the function have paramedics instead of ethics so that the second rate of double prime is the limited. H approaches zero of primal axe evaluated at X plus H three times X plus H squared minus three and there were subjecting the function minus three X squared minus three all the better for age. Next step is to simplify the numerator. Here's a binomial squared so we'LL have X squared plus two Ex age plus eight squared or not by three. Gives us three X squared class six X h plus three age squared minus three And then over here, we're going to subtract Are we're going to This should be the negatives from the ministry X squared plus three This all the h ah, now we canceled three extorted with negative three x squared And then? Then they got three and the positive three. And then agent the denominator comes around with one inch of each of the remaining terms leaving us with the limit is H approaches zero of six. Eight x plus three each and his age approaches zero three to purchase zero, leaving us with just six. X as our derivative, our secondary of F double prime of acts. So we have determined deaf prime and asked about crime I'm to recall after vexes x cubed minus tree x f Prima vex is at three x squared my street. Enough double primer vexes six x So the next part is to grab all three of these air to show that the answers seem reasonable. So the original function of X is shown in purple executed minus three x the derivative, or that it's three expert ministrations in red, and it threw her after that is six x, which is shown in blue. So the original function in purple is a cubic function. It ah is increasing up until its maximum point here, meaning that it's derivative which is shown in red must be all positive, which is true, but then as the derivative zero, meaning that service crosses the X axis. Then afterwards it's decreasing until it's minimums. All negative in that region followed a pie, and afterwards it's increasing, which means that it's rude is all positive, so that seems pretty consistent. On the drill you have prime in red. It is always decreasing up until its minimum. Here on DH. It's derivative. Here in blue is showing just that since function have prime is decreasing, double crime must be negative. And then it's ah, derivative of a derivative of brain eyes equal to zero here at X equals zero, which is true. And then afterwards it's all positive derivative, thought positive, meaning that the function of peace increasing half their X equals zero, which is true. So the graph confirms that all of these functions are consistent with each other as F f F prime and double prime.