Find the slope of the line tangent to the graph of f(x) = x / ...

Question

Find the slope of the line tangent to the graph of f(x) = x / x+6 at the point (3, 1/3) and at (-2, -1/2).

Accepted Answer

Hello. In this problem, we will be determining the slope of the tangent line at the given point of the function. So the problem gives us a function P X, which is defined as X divided by 2 X plus 8. We are also told that we want to determine the slope of the tangent line at the point 2.1 divided by 6. So how do we start this process? We'll recall that a derivative will give us the slope at any point on the original function P X. So, what we will need to do is we will need to first find the derivative of the given function P X. Notice that our function P of X is in the form of a rational function. So, in order to find the derivative, we will need to use the quotient rule. The quotient rule is defined as the following. If we want to take the derivative of a rational function, we will define the numerator and denominator as the respective functions F of X and G of X. Then the quotient rule will be defined as G of X, multiplied by the derivative F of X minus F of X multiplied by the derivative G of X, all divided by G of X squared. So, let's go ahead and define our F of X and GovX function. F of X will be the function in the numerator of the rational function, and that is going to be X. G of X will be the function in the denominator, and that will be 2 X plus 8. Now we need to take the derivatives of both the numerator and denominator. By using the power rule, the derivative of the numerator F of X will equal to 1. And by using the power rule again, G X will be defined as 2. Now, let's go ahead and take these values and plug it into the quotient rule in order to determine the derivative. So the derivative P X will be defined as G of X, which is 2 X + 8, multiplied by the derivative F of X, which is 1 minus F of X, which is negative X, multiplied by the derivative G of X, which is 2, all divided by G of X quantity squared. 2 X +8 quantity squared. Now let's go ahead and simplify our derivative. 2 X plus 8, multiplied by 1, will just give us 2 X + 8, and in negative X multiplied by 2 will give us -2 X. So the derivative will simplify to be 2 X plus 8 minus 2 X all divided by the quantity, 2 X plus 8 quantity squared. Now let's go ahead and simplify further. We can combine the terms 2 X and negative 2X, and those will cancel each other out. So the simplest form of our derivative will be 8 divided by 2 X plus 8 quantity squared. So we have a derivative that will determine the slope of the tangent line, but how are we going to get the slope? Well, in the beginning of the problem, we were told that we were interested in finding the slope specifically at the point 2.16. So in order to determine the slope at this point, we will need to use the X value of the given point. What we will do is we will take this value, which in this case is 2. And plug it into our derivative. Once we do that, that will give us the slope at the tangent of the tangent line at that required point. So, if we plug in 2 to our derivative, we will have P 2 equal to 8 divided by the quantity, 2 multiplied by 2 plus 8 squared. Now let's go ahead and simplify. 2 multiplied by 2 will give us 4, and 4 + 8 will give us the value of 12. So we have 8 divided by 12 squared. 12 squared will simplified to be 144. And the fraction 8 divided by 144, will further simplify to be 1 divided by 18, and this value is going to be the value of the slope. Of the tangent line at the point 2.16, and with that being said, the solution to this problem is going to be C. So, I hope this video helps you in understanding on how to approach this problem, and I will go ahead and see you all in the next video.

Find the slope of the line tangent to the graph of f(x) = x / x+6 at the point (3, 1/3) and at (-2, -1/2).

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