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# Normal to y=𝑒ˣ/x²

## Video transcript

so we have the function f of X is equal to e to the X over x squared and what I want to do in this video is find the equation not of the tangent line but the equation of the normal line when X is equal to 1 so we care about the equation of the normal line so I encourage you to pause this video and try this on your own and if you need a little bit of a hint the hint I will give you is that the slope of a normal line is going to be the negative reciprocal negative reciprocal of the slope of the tangent line if you imagine if you imagine a curve if you imagine a curve like this and we want to find a tangent line at a point it's going to look something like this so the tangent line is going to look like this a normal line is perpendicular to the tangent line so normal this is this this is the tangent line the normal line the normal line is going to be perpendicular to that it's going to Co just like that and if this has a slope of M then this has a slope of the negative reciprocal of M so negative 1 over m so what that is a little bit of a hint I encourage you to find the equation of the normal line to this curve when x equals 1 so let's find the let's find the slope of the tangent line and then we take the negative reciprocal we look we can find the slope of the normal line so to find the slope of the tangent line we just take the derivative here and evaluate it at x equals 1 so f prime of X and actually let me let me rewrite this a little bit so f of X is equal to e to the x times X to the negative 2 I like to rewrite it this way because I always forget the whole quotient rule thing I like the power rule a lot more and this allows me to use the power rule I'm sorry not not the power rule the product rule this allows me to do the product rule instead of the quotient rule so the derivative of this F prime of X is going to be the derivative of e to the X which is just e to the x times X to the negative 2 plus e to the x times the derivative of X to the negative 2 which is negative 2 X to the negative 3 power I just use the power rule right over here so if I want to evaluate when X is equal to 1 this is going to be equal to let me do that in that yellow color I like switching colors this is going to be equal to let's see this is going to be e to the first power which is just e times 1 to the negative 2 which is just 1 plus E Plus e to the first power which is just e times negative 2 1 to the negative 3 is just 1 so e times negative 2 so let me write it this way so top so minus 2 e and E minus 2 e is just going to be equal to negative e so this right over here this is the slope that is the slope of the tangent line the slope of the tangent and so if we want the slope of the normal we just take the negative reciprocal so the negative reciprocal of this is going to be well the reciprocal is 1 over negative even with the negative of that so it's going to be 1 over e this is going to be the slope slope of the normal line slope of normal normal line and then if we an our goal isn't just to the slope of the normal and we have the equation of the normal line and we know that the equation of a line can be represented as Y is equal to MX plus B where m is the slope so we can say it's going to be Y is equal to 1 over e remember we're doing the normal line here times X plus B and to solve for B we just have to recognize that we know a point that this goes through this goes through the point x equals 1 and when x equals 1 so when x equals 1 when x equals 1 what is y well Y is e to the first over 1 which is just e so this goes to the point 1 comma e so we know that when X is equal to 1 Y is equal to e and now we can just solve for B so we get e is equal to e is equal to 1 over e is equal to 1 over e plus B plus B or we could just subtract 1 over e from both sides and we would get B is equal to e e minus 1 over e e minus 1 over e and we could obviously write this as let's see we could write this as e squared minus 1 over e if we want to write it like that but we could just leave it just like this so the equation of the normal line so we could deserve our drumroll right over here is going to be y is equal to 1 over e 1 over e times x plus b and b plus b b is all of this so plus e minus 1 over e so that right there is our equation of the normal line