Slopes and Tangent Lines

In Exercises 13–16, ...

Question

Slopes and Tangent Lines

In Exercises 13–16, differentiate the functions and find the slope of the tangent line at the given value of the independent variable.

y = (x + 3)/(1 – x), x = −2

Accepted Answer

Hello there. Today we're going to solve the following practice problem together. So first off, let us read the problem and highlight all the key pieces of information that we need to use in order to solve this problem. Using the limit definition, find the derivative of the function and the slope of the tangent line at the given value of X. Y equals x + 6 divided by 2 minus X X equals -1. Awesome. So it appears for this particular problem, we're ultimately trying to figure out the slope of the tangent line at the given value of X, that's one of our answers we're trying to solve for, and we're also asked to find the derivative of the function. So once again, our two answers that we're ultimately trying to solve for using the limit definition is we're asked to figure out the derivative of the function and the slope of the tangent line at any given value of X, given our specific equation for Y, and we're also told that X is equal to -1. So with that in mind, now that we know we're ultimately trying to solve for, let us first off, recall and use the limit definition of the derivative. Which states that F of X is equal to the limit, as H approaches 0 of F of X plus H minus F of X, all divided by H. Awesome. So our first step is we need to determine the value of F of X plus H. So that's our first step. So we need to note that F of X plus H is going to be equal to X plus H + 6. Divided by 2 minus X plus H, which is going to be equal to X plus H + 6 divided by 2 minus X minus H. And once again, we're plugging in a value of X plus H into our expression for X on our original expression, which is Y equals X + 6 divided by 2 minus X. Awesome. So now, our second step is we need to find F of X plus H minus F of X. Fantastic. So that's our next step. So we need to take F of X plus H. Minus F of X, and this will be equal to X plus H + 6 divided by 2 minus X minus H. Minus X plus 6 divided by 2 minus X, fantastic. So now we need to find a common denominator, and when we do, we will get that F of X plus H minus F of X is going to be equal to parentheses X plus H + 6 multiplied by parentheses 2 minus X. Minus parentheses X + 6 multiplied by parentheses 2 minus X minus H. All divided by parentheses 2 minus X minus H multiplied by 2 minus X. Fantastic. So now we need to expand the numerator. That's our third step. So, when we expand the numerator, we'll get X plus H + 6 in parentheses multiplied by parentheses 2 minus X, which will be equal to 2 X plus 2 H + 12 minus X to the power of 2 minus XH minus 6 X. And we can also write that parentheses X plus 6 in parentheses multiplied by 2 minus X minus H will be equal to 2 X plus 12 minus X squad minus 6 X minus XH minus 6 H. Fantastic. So now, as we should note, the difference will be drum roll, parentheses X plus H + 6 and in parenthesis multiplied by 2 minus X minus X plus 6 multiplied by 2 minus X minus H in parentheses, which will be all equal to parentheses. 2 X plus 2 H plus 12 minus X2 minus XH minus 6 X, and don't forget your parentheses, minus 2 X plus 12 minus X squad minus 6 X minus XH minus 6 H. Fantastic. So once again, this will be all when we simplify, will be equal to ultimately 2H plus 6H, which is going to be equal to, so 2H plus 6H which is going to be equal to 8H. So now, our next step, moving right along here. As we need to for our 4th step, we need to find. Drum roll F of X plus H minus F of X, all divided by H, and we need to take the limit. So when we do that, we will find that That F of X plus H minus F of X divided by H is going to be all equal to. 8 H divided by H multiplied by 2 minus X minus H multiplied by parentheses 2 minus X. Which will be equal to, when we simplify, 8 divided by 2 minus X minus H multiplied by, and don't forget that 2 minus X minus H is in parenthesis multiplied by 2 minus X. Fantastic. So now if we take the limit as H approaches 0, so this once again, this is as H approaches 0, we could go ahead and write that F of X is going to be equal to 8 divided by parentheses 2 minus X2. Or Y of X is gonna be equal to 8 divided by parentheses 2 minus X to the power of 2. Fantastic. So that Is one of our final answers. Hooray, we did it. So now, our next step is we need to calculate the slope of the tangent line at X equals -1. So we need to recall that the slope of the line tangent to the curve at X equals -1 is, so let's write this down. So once again, as we should recall, the slope of the tangent line. To the curve at X equals -1 will be Y of -1. So that will mean that we need to substitute in so we need to substitute in the value of X equals -1 into our derivative. So when we do that, we will find that Y. of -1 is gonna be equal to 8 divided by 2 minus -1 to the 2, which when we plug that into our calculator, we will find that our And final answer will have to be 8 divided by 9 or 8 9th, so 8 divided by 9. So therefore, without a doubt, Y-1 is going to be equal to 8 divided by 9, and that is our final answer for the second answer we were trying to solve for. Hooray, we did it. So going back up to the top to quickly recap what we've learned for this particular problem. Without a doubt, our final two answers have to be as follows. So Y of X has to be X has to be equal to 8 divided by parentheses 2 minus x 2. And Y of -1 has to be equal to 8 divided by 9 or 89. Thank you so much for watching. Hopefully that helped and I can't wait to see you in the next video. Bye.

Slopes and Tangent Lines

In Exercises 13–16, differentiate the functions and find the slope of the tangent line at the given value of the independent variable.

y = (x + 3)/(1 – x), x = −2

Key Concepts

Differentiation

Quotient Rule

Slope of the Tangent Line

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