# Question Video: Finding the Value of the Derivative of a Function at a Point given the Integration of the Function Mathematics • Higher Education

Given that β« π(π₯) dπ₯ = π₯Β³ β 7π₯Β² β π₯ + 9 + πΆ, find πβ²(1).

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### Video Transcript

Given that the integral of π of π₯ with respect π₯ is equal to π₯ cubed minus seven π₯ squared minus π₯ plus nine plus πΆ, find π prime of one.

In this question, weβve been told that the integral of some function is equal to some polynomial. To be able to find the first derivative of the function evaluated when π₯ is equal to one, thatβs π prime of one, weβll begin by using the fundamental theorem of calculus to simply find an expression for the function itself. And of course, this theorem links the concept of differentiating a function with the concept of integrating a function. We can find a function π of π₯ by differentiating the function π₯ cubed minus seven π₯ squared minus π₯ plus nine plus πΆ with respect to π₯. And Iβm going to rewrite this using Leibnizβs notation to prevent any confusion.

Here, we recall the fact that the derivative of the sum of a number of terms is equal to the sum of the derivative of each of those terms. So, we see that π of π₯ is equal to the derivative of π₯ cubed with respect to π₯ plus the derivative of negative seven π₯ squared with respect to π₯, and so on. We then recall that the derivative of a general polynomial term, ππ₯ to the power of π, for real constants π and π is π times π times π₯ to the power of π minus one. In other words, we multiply by the exponent and then reduce that exponent by one. And this tells us that the derivative of π₯ cubed is three π₯ squared. We then obtain the derivative of negative seven π₯ squared to be two times negative seven π₯ to the power of one, which is simply negative 14π₯.

Another way of saying negative π₯ is saying negative π₯ to the power of one. We multiply by one and then reduce that power by one. So, that gives us one times negative π₯ to the power of zero. But of course, π₯ to the power of zero is one. So, we see that the derivative of negative π₯ is negative one. And we distribute the parentheses because we donβt really need them here. The derivative of a constant is, of course, zero. So, the derivative of nine and the derivative of πΆ with respect to π₯ is zero. And we found π of π₯. Itβs three π₯ squared minus 14π₯ minus one.

Weβre not quite done, though. In this question, weβre looking to find the first derivative of our function and then evaluate it when π₯ is equal to one. So, we use a similar process to differentiate π of π₯ with respect to π₯. The derivative of three π₯ squared is two times three π₯ to the power of one, which is six π₯. The derivative of negative 14π₯ is negative 14. And of course, the derivative of negative one is zero. So, π prime of π₯ is equal to six π₯ minus 14. And we can simply replace π₯ with one to find π prime of one. Thatβs six times one minus 14, which is, of course, equal to negative eight. Given the definition of our function π of π₯, then π prime of one is negative eight.