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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}} $

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

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Campbell University

Oregon State University

Baylor University

Idaho State University

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.

Oregon State University