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

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 1 / AB

Chapter 3

DIFFERENTIATION

Section 5

Higher Derivatives

Derivatives

Differentiation

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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'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'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're 15 or 16. Name seven over too fit through the banks again. We'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's always gonna be some multiple because there'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's positive and the even during derivatives it's negative. And if we look at this rule native, want the and minus one, it'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't play the odd numbers 13 and five as the numerator and the denominator is to to power. And so that'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.

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