Explain how we can use the Fundamental Theorem of Calculus (part I) to calculate the derivative of the function: $f(x) = \int {(((\dfrac{1}{3}){t^2})} - 1{)^7}dt$ from $x$ to $3$?

Hint: Here the question has asked to find derivation for an integral, so it would be a waste of time to evaluate that integral, rather to get the answer we must directly use the fundamental theorem of calculus. The theorem says: $\dfrac{d}{{dx}}\int\limits_a^x {f(t)dt = f(x)} $; where a is some constant. Naturally it says that the derivative of an integral means the function itself. Remember this theorem to solve this problem in minutes.

Complete step by step answer:
First of all let us write down the given data to understand the question better;
The function given to us is: $f(x) = \int {(((\dfrac{1}{3}){t^2})} - 1{)^7}dt$; if we analyze this function, we can see that there is an integral within the function, and the question has asked us to find the derivative of this function.
Now if we look at the question and proceed to integrate it, we will be wasting our time. This is because; there is a pattern in the question. The function has an integral in it and we are asked to find its derivative, we have a theorem available which can solve questions of this pattern. Let us remember that the Fundamental theorem of calculus states that;
$\dfrac{d}{{dx}}\int\limits_a^x {f(t)dt = f(x)} $; where a is some constant
According to the question;
\[f(x) = \int\limits_x^3 {(((\dfrac{1}{3}){t^2} - 1)} {)^7}dt\]

Now what we require is;
$f(x) = \dfrac{d}{{dx}}\int\limits_x^3 {{{((\dfrac{1}{3}){t^2} - 1)}^7}dt} $
In order for us to get the right form of the function, we have to swap the limits;
$f'(x) = \dfrac{d}{{dx}}( - \int\limits_3^x {{{((\dfrac{1}{3}){t^2} - 1)}^7}dt} )$
$ \Rightarrow f'(x) = - \dfrac{d}{{dx}}(\int\limits_3^x {{{((\dfrac{1}{3}){t^2} - 1)}^7}dt} )$
We can now go to our final step, that is applying the Fundamental theorem of calculus;
$ \Rightarrow f'(x) = - {((\dfrac{1}{3}){x^2} - 1)^7}$

$\therefore $ The final answer is $f'(x) = - {((\dfrac{1}{3}){x^2} - 1)^7}$.

Note: While studying calculus we need to remember that it comprises some important and interrelated concepts. The concepts that we utilize in calculus are: limits, functions, integrals, derivatives, also infinite series. We need calculus in various works of life that is its applications range from economics to engineering. There are two fundamental theorems used in calculus, one is for a single constant and the other is for a pair of constants.