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(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 $

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

00:45

Amrita Bhasin

Calculus 1 / AB

Chapter 3

Differentiation Rules

Section 10

Linear Approximation and Differentials

Derivatives

Differentiation

Missouri State University

Harvey Mudd College

Idaho State University

Lectures

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In mathematics, a derivative is a measure of how a function changes as its input changes. Loosely speaking, a derivative can be thought of as how much one quantity is changing in response to changes in some other quantity; for example, the derivative of the position of a moving object with respect to time is the object's velocity. The concept of a derivative developed as a way to measure the steepness of a curve; the concept was ultimately generalized and now "derivative" is often used to refer to the relationship between two variables, independent and dependent, and to various related notions, such as the differential.

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In mathematics, a differentiation rule is a rule for computing the derivative of a function in one variable. Many differentiation rules can be expressed as a product rule.

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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.

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