`y = sqrt(2 - e^x)` 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/

Textbook Question

Chapter 3, 3.4 - Problem 6 - Calculus: Early Transcendentals (7th Edition, James Stewart).
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hkj1385's profile pic

hkj1385 | (Level 1) Assistant Educator

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Note:- 1) If y = sqrt(nx) ; then dy/dx = n/[2*sqrt(nx)]

Now, 

y = sqrt{2 - (e^x)}

Let  g(x) = sqrt(x) ..............(outer function)

and f(x) = {2 - (e^x)}........(inner function)

Thus, g(f(x)) = sqrt{2 - (e^x)}

Now, dy/dx = y' = [1/2*sqrt{2 - (e^x)}]*[-e^(x)]

or, dy/dx = -(e^x)/[{2*sqrt{2 - (e^x)}}]

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balajia | College Teacher | (Level 1) eNoter

Posted on

The given function is `y=sqrt(2-e^x)`

This is in the form `y=f(g(x))`

Here `f(x)=sqrt(x)` and `g(x)=2-e^x`

`y'=(1/(2sqrt(2-e^x))).(-e^x)`

`y'=(-e^x)/(2sqrt(2-e^x))`

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