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Question

Calculate the first five derivatives of $f(x)=\sqrt{x}$
$$
\begin{array}{l}{	ext { (a) Show that } f^{(n)}(x) 	ext { is a multiple of } x^{-n+1 / 2}} \ {	ext { (b) Show that } f^{(n)}(x) 	ext { alternates in sign as }(-1)^{n-1} 	ext { for } n \geq 1 	ext { . }} \ {	ext { (c) Find a formula for } f^{(n)}(x) 	ext { for } n \geq 2 . 	ext { Hint: Verify that the coefficient }} \ {	ext {  is } \pm 1 \cdot 3 \cdot 5 \cdots \frac{2 n-3}{2^{n}}}\end{array}
$$

Yogesh Sawant · Accepted Answer

okay. Here were given a function half of X equals a square root of X. No, if it actually goes the square of X and we have to find first five derivatives, Okay. And we&#x27;re given these three instructions showing that the end due to FX the multiple of actual power negative n plus 1/2 showing that the sign alternates and finding in general formula. So first of all, it&#x27;s beneficial to rewrite this as actually power of 1/2 so that we can use the power rule. Just get started. First review of X Using a parole is going to be 1/2 X to the negative 1/2 because against quick refresher, actually part of and the derivative becomes an ex parrot and minus one. So again, second derivative, we got most of this to get negative one Force X to the negative three halves. What? Third room X. Again, we use the power room. Multiply these two fractions. Now this one becomes positive again because the two negatives cancel yet 3/8 x donated five halves. Fourth derivative rex. And then we multiply fractions. You&#x27;re 15 or 16. Name seven over too fit through the banks again. We&#x27;ll plant fraction Cornu power rule. Get 105 over 32 terms. Ex Canadian, nine halves. And this one here should be negative. Just forgot, native son there because when you multiply fractions here. All right. So I followed the instructions. We calculate the 1st 5 derivatives, and now I have to show that the Institute of X is a multiple acts the part and the minus 1/2 So in this case, and equals one, two, 34 and five. So for X if n is one x the power of negative one plus 1/2 native one plus 1/2 is nave. One has and we can try that again. Here, native to close 1/2 equals negative three house. As you can see, the pattern is going to continue, sir. We know the ex that the end through there&#x27;s always gonna be some multiple because there&#x27;s these fractions. But the X values also be raised to native and plus 1/2. And now for the sign, we can see the alternates between positive and negative. And we know that for the aw derivatives 13 and five, it&#x27;s positive and the even during derivatives it&#x27;s negative. And if we look at this rule native, want the and minus one, it&#x27;s going to show that the drift is going to be native or positive. So alternates inside. Finally, we can put it all together in the general formula for the anthem narrative and were given this alternating. You want the caravan, my sworn And then we have the fraction that we can verify here. If we add up, the odd we won&#x27;t play the odd numbers 13 and five as the numerator and the denominator is to to power. And so that&#x27;s why we can say one times three times five to analyze. Three over to to Fire of End. And this is extra power of needed and plus one, yeah.

Amy Jiang · Answer

okay, So protective. You&#x27;re a bit of a breath of xn G of X. Well, we use our peripheral. That&#x27;s f prime of that Times Geographics. And then plus G time of X Times effort back. And this is that&#x27;s accidentally one talking at a table when accident. One halftime of one with a drink. Guillette one. That&#x27;s, um or do you prime at one. That&#x27;s two in that one. That&#x27;s for it. So what do we get? We get 12 plus 10 and that&#x27;s equal to 22.

Amy Jiang · Answer

Okay, so let&#x27;s take the derivative of the following. So we use our product roll derivative of X. That&#x27;s this one times f of X plus the derivative of off time sex. And this is all excellently today. So what does that give us looking at our table off at pre that people to three plus of prime at three? That&#x27;s too time Jax, which is three. So that&#x27;s six close three, which is equal to nine.

Anna Waldram · Answer

given this table, we want to find the derivative of FX plus g of X when X equals one. So we&#x27;ll start off using these some rule. So we&#x27;ll take the some of the derivatives of the two functions of prime of X plus G Prime of X, where X equals one, and now we can go ahead and plug in one. So that&#x27;s the derivative of F prime of one less G, prime of one. And now we can take a look at this chart and it tells us that f prime of one is three and g prime of one is to. So we can just substitute those right and we get three plus two equals the derivative, and that equals five, and that is our answer.

Ernest Castorena · Answer

the function of a Rex is equals x to the 10th minus three x to the fourth plus two x minus on. And then they want us to compute the fifth derivative. But know that the fifth derivative rules they were out any polynomial. Let&#x27;s degree for less so we won&#x27;t care about the rest. Take our derivatives. Let&#x27;s see, the first derivative will give us 10 x to the ninth will be some stuff over. Alright, second derivative. Well, give us but seeing 90 x today. Some of us over right there, derivative. Well, give us seeing 90 times eight next to you seven means I&#x27;m not covered. And Lynsey, our fourth derivative Willie, was my district time. Sometimes nothing. Be time. Seven Thanks to the six one term leftover Probably lineup. Now 50. It&#x27;s the only time left over. Well, in the book. 10 times nine times 8 10 76 X to the fifth and the rest of the terms. Evans, Right out. So this is our derivative doesn&#x27;t

Alex Johnson · Answer

and this problem, we&#x27;re gonna calculate the derivative of F of X and instead of using the product rule a couple times, let&#x27;s work on distributing the parentheses and simplifying. Ffx weakens the power rule so we have X squared minus X plus three x minus three multiplied by the corn. The expert is five, which gives us X squared plus two x minus three times quantity X minus five Ellen. Let&#x27;s distribute again So we have X cubed minus five X squared plus two X squared minus 10 X minus three X plus 15 and this simplifies two X cubed minus three X squared minus 13 x plus 15. Okay, now that we have ever vex simplified, let&#x27;s use the power rule and calculate f prime vex. This gives us three X squared minus six x minus 13.

Problem

In Exercises $31-36,$ find a general formula for …

Problem 30 Medium Difficulty

Answer

a)$$
x^{-n+\frac{1}{2}}
$$
b) $$
f^{(n)}(x) \text { is }(-1)^{n-1} \text { for } n \geq 1
$$
c) $$
f^{(n)}(x)=\frac{(-1)^{n-1} 1.3 .5 \ldots(2 n-3)}{2^{n}} x^{-n+\frac{1}{2}}
$$

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a)$$x^{-n+\frac{1}{2}}$$b) $$f^{(n)}(x) \text { is }(-1)^{n-1} \text { for } n \geq 1$$c) $$f^{(n)}(x)=\frac{(-1)^{n-1} 1.3 .5 \ldots(2 n-3)}{2^{n}} x^{-n+\frac{1}{2}}$$

Calculus for AP

Catherine R.

Caleb E.

Samuel H.

Joseph L.

Precalculus Review - Intro

Functions on the Real Line - Overview

Jon Rogawski & Ray CannonCalculus for AP

Catherine R.

Caleb E.

Samuel H.

Joseph L.

a)$$
x^{-n+\frac{1}{2}}
$$
b) $$
f^{(n)}(x) \text { is }(-1)^{n-1} \text { for } n \geq 1
$$
c) $$
f^{(n)}(x)=\frac{(-1)^{n-1} 1.3 .5 \ldots(2 n-3)}{2^{n}} x^{-n+\frac{1}{2}}
$$

Jon Rogawski & Ray Cannon

Calculus for AP