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

[maximus101]

Homework Statement

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

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

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

Homework Equations

The Attempt at a Solution

would like to work through this with person/people please. Not sure were to start

 

[Char. Limit]

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.

 

[jaci55555]

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?

 

[Char. Limit]

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.