### Video Transcript

The slope at the point π₯, π¦ on the graph of a function is five π₯ minus two over π₯. Find the equation of the curve if it contains the point π, five π plus three.

To answer this question, itβs first useful to notice that weβve been given an expression for the slope of our function. Now, we know that the slope at any point is found by differentiating the expression for the original function with respect to π₯. So if weβre given a curve π¦ is equal to some function of π₯, then the derivative dπ¦ by dπ₯ is its slope; itβs a given point. So here, dπ¦ by dπ₯ is equal to five π₯ minus two over π₯.

Now, the fundamental theorem of calculus tells us that differentiation and integration are essentially opposite operations. And so if we integrate our expression for dπ¦ by dπ₯ with respect to π₯, weβll get our equation for π¦. Now, in fact, this will be a general equation. Weβll need to use the fact that our curve passes through the point π, five π plus three to find a particular equation. So letβs begin by integrating dπ¦ by dπ₯; thatβs five π₯ minus two over π₯ with respect to π₯. Now, before we perform any integration, we should simplify our integrand. Weβll simplify by dividing each term in the numerator by π₯. So π¦ is equal to the indefinite integral of five minus two over π₯ with respect to π₯.

Now, letβs integrate term by term. The integral of five is five π₯. Then, the integral of one over π₯ is the natural log of the absolute value of π₯. So the integral of negative two over π₯ is negative two times the natural log of the absolute value of π₯. Now, of course, weβre working with an indefinite integral, so we need that constant of integration. So we find π¦ is equal to five π₯ minus two times the natural log of the absolute value of π₯ plus πΆ.

But of course, to find the equation of our curve, we need to work out what πΆ is. Well, we were told that our curve passes through the point π, five π plus three. In other words, when π₯ is equal to π, π¦ is equal to five π plus three. So letβs substitute these values in. We get five π plus three equals five π minus two times the natural log of the absolute value of π plus πΆ. Now, we know that π is itself a positive number, so we donβt need these absolute value signs. We also know that the natural log of π is simply one. So negative two times the natural log of π is negative two. And our equation is now five π plus three equals five π minus two plus πΆ. We can subtract five π from both sides. Then we add two. And we find πΆ is equal to five.

And so we found the equation of our curve given information about its slope and a point that it passes through. Itβs π¦ equals five π₯ minus two times the natural log of the absolute value of π₯ plus five.