How do you use the fundamental theorem of calculus to find F'(x) given #F(x)=int 1/t^2dt# from [1,x]?

1 Answer
Mar 22, 2018

Answer:

#F'(x)=1/x^2#

Explanation:

The first part of the Fundamental Theorem of Calculus tells us that if

#F(x)=int_a^xf(t)dt# where #a# is just a constant, then #F'(x)=f(x)#.

This makes sense -- #F(x)# is an integral, differentiating #F(x)# to get #F'(x)# should just give us what we originally integrated, IE, the integrand #F(t)# evaluated from a constant to #x#.

The most important thing to note is that the variable of the derivative and variable of the integrand are different. The integrand is written in terms of #t,# the integral and derivative in terms of #x.#

Here, we have

#F(x)=int_1^x1/t^2dt#

And we see #a=1# (the value of #a# is totally irrelevant to our final answer, noting it anyways), #f(t)=1/t^2.#

Thus,

#F'(x)=1/x^2#