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Problem 15 Easy Difficulty

(a) Find the differential $ dy $ and (b) evaluate $ dy $ for the given values of $ x $ and $ dx. $
$ y = e^{x/10}, x = 0, dx = 0.1 $

Answer

(a)$\frac{1}{10} e^{x / 10} d x$
(b) $$0.01$$

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Video Transcript

Okay in this problem we have Y equals E. To the X over 10 power. And uh we want to find the differential dy and uh evaluated uh for this value effects and this value of dx Well defined. Dy we have to find dy dx the derivative. No, we want to find a derivative uh of uh why Y equals E to the X over 10. So we need to find dy dx. Now, what is the derivative of E to the X over 10? You can think of X over 10 as let's write it down. You can think of X over 10 As 1/10 times x. So you can think of this as E to the 1/10 times X power. Same thing as E to the X over 10 power. If you want to take the derivative Of E to the 1/10 x. The derivative of E T. D. You is easily you and then we have to take the derivative of U with respect to X. We're using the chain rule. So derivative of uh if this was you derivative of U with respect to X derivative of 1/10 times X is just 1/10. So dy dx is really equal to 1/10 Times E. to the 1/10 x. Or you can rewrite this as X over 10 if you wanted to. But that is our derivative. Now, if dy dx is equal to this, then multiplying both sides by D X will give us our differential Dy because D X divided by and dx being times will cancel. So we have our differential in why is equal to 1/10 times E. Teddy. X over 10. 1 10 times X. We can rewrite as X over 10 time's D. Yet. So here is our differential dy uh huh. Now if we want to evaluate uh dy for this particular X we're gonna plug zero in for X. And for this particular D. X two differential of X. Dx is going to be 20.1 0.1 will replace D. X. Zero will place X. So D. Y. Our differential of why uh for this value X. And this value dx dy equals 1/10 times E. To the X over 10 which will be 0/10. Which of course will of course be zero. Yeah. Uh So 1/10 times he to the 0/10 since X zero times D. X. Which is going to be 00.1. No ah 0/10 zero E. Raised to the zero power. Anything raised to the zero power is one. So D Y is going to be 1/10 times one Times 1 10.1 is 1/10. So D. Y equals 1/10 times one times 1/10 which is 100. Or we could just write it as 1000.1. So our expression for the differential Y dy equals 1/10 times E. To the X over 10 D. X. And our differential. And Y Dy evaluated for this x value and this DX value is .01.