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, is that the

slope of a normal line is going to be the

negative reciprocal of the slope of

the tangent line. 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. This is the tangent line. The normal line is going to

be perpendicular to that. It’s going to go 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/m. So with that as 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 slope

of the tangent line. And then we take the

negative reciprocal, 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 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 the power

rule, the product rule. So 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 2x to

the negative 3 power. I just used 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 like. 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 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 minus 2e. And e minus 2e is just going

to be equal to negative e. So this right over here, this is

the slope of the tangent line. 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 e, but we want the

negative of that. So it’s going to be 1/e. This is going to be the

slope of the normal line. And then if we, and

our goal isn’t just to the slope of

the normal line, we want the equation

of the normal line. And we know 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/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, what is y? Well, y is e to the 1st

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 1/e plus b. Or we could just subtract

1 over e from both sides, and we would get b is

equal to e minus 1/e. And we could obviously right

this as e squared minus 1/e if we want to

write it like that. But could just leave

it just like this. So the equation of

the normal line– so we deserve our drum

roll right over here– is going to be y is equal

to 1/e times x, plus b. And b, plus b, is all of this. So plus e minus 1/e. So that right there is our

equation of the normal line.

I read Equation of Normal Life 😛

Lost me really quickly

i have question. for the eq.

x^2+y^2=(2X^2+2y^2-x)^2

@ point (1/2,0)? the y'=-1/0. what is the tangent and normal line? pls. reply sir. thanks☺.

That was a good one……

ｗｏｋｅ

Math is magic step.Thanks a lot .

watch the video at speed 1.25x or even 1.5x if you can, much better.

why normal line is negative slope?

well i failed

can someone tell me why do we need to find the normal and tangent lines? why do we need them, as in the purpose. Thank you.

thanks I understood

Luv you long time Khan. Saved my life many of times

Oh! So the normal line is perpendicular to the tangent line. Thank you so much for your help!