Home > Calculus > The Fundamental Theorem of Calculus

How do you find #F'(x)# given #F(x)=int 1/t dt# from #[1,x]#?

Answer 1

Paisley Johnson

# F'(x) = 1/x #

We apply the First Fundamental theorem of calculus which states that if (where #a# is constant).

# F(x) = int_a^x f(t)\ dt. #

Then:

# F'(x) = f(x)#

(ie the derivative of an anti-derivative of a function is the function you started with)

So if we have

# F(x) = int_1^x 1/t \ dt #

Then

# F'(x) = 1/x #

By signing up, you agree to our Terms of Service and Privacy Policy

Answer 2

Elijah Abram

To find ( F'(x) ) given ( F(x) = \int_{1}^{x} \frac{1}{t} , dt ), you apply the Fundamental Theorem of Calculus. According to this theorem, if ( F(x) ) is a function defined by an integral ( F(x) = \int_{a}^{x} f(t) , dt ), where ( f(t) ) is a continuous function on the interval ([a, x]), then ( F'(x) = f(x) ).

In this case, ( F(x) ) is already given as ( F(x) = \int_{1}^{x} \frac{1}{t} , dt ). So, to find ( F'(x) ), you differentiate ( F(x) ) with respect to ( x ):

[ F'(x) = \frac{d}{dx} \left( \int_{1}^{x} \frac{1}{t} , dt \right) ]

By applying the Fundamental Theorem of Calculus, ( F'(x) = \frac{1}{x} ).

By signing up, you agree to our Terms of Service and Privacy Policy

Answer from HIX Tutor

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.