`h(x) = int_1^(e^x)(ln(t))dt` Use Part 1 of the Fundamental Theorem of Calculus to find the derivative of the function.

Textbook Question

Chapter 5, 5.3 - Problem 13 - Calculus: Early Transcendentals (7th Edition, James Stewart).
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Borys Shumyatskiy | College Teacher | (Level 3) Associate Educator

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Hello!

Part 1 of the Fundamental Theorem of Calculus states that for a continuous function `f`

`F'_a(x)=f(x),` where `F_a(x)=int_a^xf(t)dt.`

 

Here `f(t)=ln(t)` and `h(x)=F_1(e^x).`

 

Therefore

`h'(x)=d/(dx)(h(x))=d/(dx)(F_1(e^x))=F'_1(e^x)*e^x=f(e^x)*e^x=`

`=ln(e^x)*e^x=xe^x.`

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