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# The derivative & tangent line equations

AP.CALC:
CHA‑2 (EU)
,
CHA‑2.B (LO)
,
CHA‑2.B.3 (EK)
,
CHA‑2.B.4 (EK)
,
CHA‑2.C (LO)
,
CHA‑2.C.1 (EK)

## Video transcript

we're told that the tangent line to the graph of function at the point two comma three passes through the point seven comma six find f prime of two so whenever you see something like this it doesn't hurt to try to visualize it you might want to draw it out or just visualize it in your head but since you can't get in my head I will draw it out so let me draw the information that they are giving us so that's x axis that is the y axis let's see the relevant points here two comma three and seven comma six so let me go one two three four five six seven along the x axis and I'm going to go one two three four five and six along the y axis and now this point so we have the point 2 comma 3 so let me mark that so 2 comma 3 is right over there so it's 2 comma 3 and we also have the point 7 comma 6 7 comma 6 is going to be right over there 7 comma 6 now let's remind ourselves what they're saying they sing the tangent line to the graph of function f at this point passes through the point 7 comma 6 so if it's the tangent line to the graph at that point it must go through 2 comma 3 that's the only place where it intersects our graph and it goes through 7 comma 6 so you only need two points to define a line and so the tangent line is going to look like it's going to look like let me see if I can that's not right let me draw it like it's going to look and that's not exactly right let me try it one more time okay there you go so the tangent line is going to look like that it goes it's tangent to f right at 2 comma 3 and it goes through the point 7 comma 6 and so we don't know anything other than F but we can imagine what F looks like our function f could so our function f it could look something like this it just has to be tangent so that line is to be tangent to our function right at that point so our function f could look something like that so when they say find f prime of two they're really saying what is the slope of the tangent line when X is equal to two so when X is equal to two well the slope of the tangent line is the slope of this line they gave us they gave us the two points that sit on the tangent line so we just have to figure out its slope because that is going to be the rate of change of that function right over there it's derivative it's going to be the slope of the tangent line because this is the tangent line so let's do that so as we know slope is change in Y over change in X so if we change our to go from two comma three to seven comma six our X change in X we go from x equals 2 to x equals seven so our change in X is equal to five and our change in Y our change in Y we go from y equals three to y equals six so our change in Y is equal to three so our change in Y over change in X is going to be three over five which is the slope of this line which is the derivative of the function at two because this is the tangent line at x equals two let's do another one of these for a function G we are given the G of negative one equals three and G prime of negative one is equal to negative two what is the equation of the tangent line to the graph of G at x equals negative one all right so once again I think it will be helpful to graph this so we have our y-axis we have our x-axis and let's see we say for function G we are given that G of negative one is equal to three so the point negative one comma three is on our function this is negative one and then we have one two and three so that's that right over there that is the point that is the point negative one comma three it's going to be on our function and we also know that G prime of negative one is equal to negative two so the slope of the tangent line right at that point on our function is going to be negative two that's what that tells us the slope of the tangent line when X is equal to negative one is equal to negative two so I could use that information to actually draw the tangent line so let me see if I can let me see if I can do this so it will look so I think it will let me just draw it like this so it's going to go so that's a slope of negative two is going to look something like that so as we can see if we move positive one in the X direction we go down two in the Y direction so that has a slope of negative two and so you might say well where is G well we could draw what G could look like G might look something like this might look something like that right over there where that is the the tangent line we can make G do all sorts of crazy things after that but all we really care about is the equation for this green line and there's a couple of ways that you could do this you could say well look a line is generally there's a bunch of different ways where you can define the equation for a line you could say a line has a form Y is equal to MX plus B where m is the slope and B is the y-intercept well we already know what the slope of this line is it is negative two so we could say Y is equal to negative two negative two times X times X plus B and then to solve for B we know that the point negative one comma three is on this line and this goes back to some of your Algebra one that you might have learned a few years ago so let's substitute negative one and three efore x and y so when y is equal to 3 so 3 3 is equal to is equal to negative 2 negative 2 times x times negative 1 times negative 1 plus b plus b and so let's see this is negative 2 times negative 1 is positive 2 and so if you subtract 2 from both sides you get 1 is equal to b and there you have it that is the equation of our line y is equal to negative 2x plus 1 and there's other ways that you could have done this you could have written the line in point-slope form or you could have done it this way you could have written in standard form but at least this is the way my brain likes to process it