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Write the composite function in the form $ f(g(x)). $ [Identify the inner function $ u = g(x) $ and the outer function $ y = f(u). $ ] Then find the derivative $ dy/ dx. $$ y = e^{\sqrt{x}} $

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$=\frac{e^{\sqrt{x}}}{2 \sqrt{x}}$

00:22

Frank Lin

Calculus 1 / AB

Chapter 3

Differentiation Rules

Section 4

The Chain Rule

Derivatives

Differentiation

Missouri State University

Campbell University

Harvey Mudd College

University of Nottingham

Lectures

04:40

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.

44:57

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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Write the composite functi…

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write the composite functi…

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Write the composite func…

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03:04

here we have a composite function and we're going to identify the inside function and the outside function before we differentiate. So the inside function would be the square root of X that is inside the E to the X function. And I would like to write the square root of X as extra the 1/2 when I differentiate it. So then the outside function is f of X equals e to the X okay, to find the derivative Do I d. X? What we want to dio according to the chain rule is start by taking the derivative of the outside function and we have learned that the derivative of each of the X is e to the X, So the derivative of each of the square root X we're going to have to start with the to the square root X. Then we multiply by the derivative of the inside function. So now we're taking the derivative of X to the 1/2 power and that would be 1/2 times X to the negative 1/2 power just using the power rule. Okay, so we have our derivative and now we're going to simplify So one of the things we can do is eliminate the negative exponents. So we have e to the square root X times 1/2 x to the 1/2 because X to the negative 1/2 is equivalent to one over X to the 1/2 and now we can just write it as a single fraction. So we have e to the square root X over two X to the 1/2. But why don't we take it one step further and change back the X to the 1/2 into radical notation and we have each of the square root x over two square root X that's are derivative.

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