Find y'(x) and y''(x) when y is defined with an integral

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Homework Help Overview

The problem involves differentiating a function defined by an integral, specifically calculating the first and second derivatives of y(x) which is expressed as an integral involving the square root of a function of e raised to a variable.

Discussion Character

  • Exploratory, Conceptual clarification, Mathematical reasoning

Approaches and Questions Raised

  • Participants discuss the application of the Fundamental Theorem of Calculus (FTC) and the product rule in differentiating the integral expression. There is uncertainty about whether the integral needs to be solved first and questions arise regarding the nature of the FTC.

Discussion Status

The discussion is active with participants exploring different approaches to the differentiation process. Some guidance has been offered regarding the use of the FTC and product rule, but there is no consensus on the necessity of solving the integral beforehand.

Contextual Notes

Participants are navigating through the differentiation of an integral expression, with some expressing uncertainty about the steps required and the implications of the FTC.

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Homework Statement



For x > 0, define [tex]y(x) := x[/tex] [tex]\int_0^{log x} \! \sqrt{1 + e^t} dt -(2/3)(1 + x)^{3/2}[/tex]

Calculate [tex]y'(x) := dy/dx[/tex]
and

y''(x) := [tex]d^2y/dx^2[/tex]

Homework Equations





The Attempt at a Solution



would like to work through this with person/people please. Not sure were to start
 
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Firstly, the FTC and the product rule will help you with that x*I(x). The second part of the function is easy to differentiate.
 
Char. Limit said:
Firstly, the FTC and the product rule will help you with that x*I(x). The second part of the function is easy to differentiate.

what is the FTC? will he first need to solve the integral?
 
The FTC is the Fundamental Theorem of Calculus. You could solve the integral, I suppose, but it would involve two substitutions, the first being u=e^t.
 

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