# How to Find the Equation of a Tangent Line with Derivatives (NancyPi)

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Published on May 15, 2018 by

MIT grad shows how to find the tangent line equation using a derivative (Calculus). To skip ahead: 1) For a BASIC example, skip to time 0:44. 2) For an example that uses the QUOTIENT RULE, skip to time 5:58. 3) For an example that uses TRIG FUNCTIONS and the CHAIN RULE for finding the equation of a tangent line, skip to time 11:35. Nancy formerly of MathBFF explains the steps.

What is a tangent line? It is a line that is tangent to a curve at one point. You can use calculus for finding the tangent line equation, by taking the derivative of the given function.

Finding the EQUATION OF A TANGENT LINE using a DERIVATIVE:
There are four main steps to find the equation of a tangent line with derivatives:

1) TAKE DERIVATIVE: The first step is to take the derivative of the given equation, with respect to x. For instance, to find the equation of the line tangent to f(x) = x^2 + 3 at x = 4, first take the derivative of f(x), which is 2x, by the power rule.

2) PLUG IN X-VALUE INTO DERIVATIVE TO GET SLOPE: The second step is to plug the given x-value into the derivative of f(x). The value you get is the slope of the tangent line, m.

3) FIND Y-VALUE OF POINT WITH ORIGINAL EQUATION: The third step is to find the y-value of the point. You get this by plugging the given x-value into the original equation to find the corresponding y-value. Since you want the tangent line at x = 4, plug x = 4 into the original f(x), and you get a y-value of 7. Now that you have both the x and y coordinates, this means that the full point is (4,7).

4) PLUG THE SLOPE AND X,Y POINT VALUES INTO POINT-SLOPE EQUATION: The last step is to put the x and y point values, as well as the slope, m, into the point slope equation of a line. The point slope form is x - x1 = m (y - y1). Plug the x and y point values into this equation for x1 and y1. Plug the slope you found in for m. This is the equation of the tangent line. The equation will still have x and y variables in it.

You can also rearrange the equation you get so that it is in "y equals" or slope intercept form, if you want.

You may need to find the derivative with other derivative rules, such as the quotient rule or chain rule. Even if you need to use other rules when taking derivatives, or even if you need to differentiate something like a trigonometry function (ex. secant) or a fraction form with a numerator and denominator (rational function), the four main steps to find the equation of the tangent line will be the same.

For more math help, check out: http://nancypi.com

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• Welcome back!

• Love you! Keep up the good work! 🙂

• 1

• hey nancy, can you make a video on solving optimization problems using calculus?

• Yes, videos on complicated and harder calculus problems?

• Yes! optimization or related rates..

• whew, I still remember calculus

• For me it is simpler to consider that there are an infinite number of lines with a given slope (say, 5). That is, lines that satisfy:
y = 5x + b
But only one contains a given point [say (1,3)]:
3 = 5*1 + b
Solving for b give -2, so we know the equation of the tangent line is
y = 5x – 2
I think this method gets students ready for initial value problems and avoids the symbol overload of the 5 variables in the point-slope equation.

• you’re a god

• Could u do one with horizontal line

• how did you get cos(pi/3) is equals to 1/2 and tan(pi/3) is equals to square root of 3?

• 2:47 i can’t understand why the slope m is a result from the x-coordinate only?

• I am the 101 like 😉 appreciable. very nice explanation

• Nice

• 2:52 “The third step is…oh never mind. Fuck you guys.”

• thank u so much madam…

• hey Nancy thanks a lot now I can easily solve tangent

• how do you write on that i’m perplexed

• @YoIts LemonBoy! Therefore, she is not left handed so she may have a wedding ring on. Bummers.

• @Hugo Anzola why are people like you so creepy. dude go outside and meet someone or soemthing. stop tryna marry a tutor online.

• Hugo Anzola I’m pretty sure she’s left handed

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• @Diego Villarroel she’s right handed.
Or she’s left handed and has mastered writing backwards.
What is most likely?
Bear in mind about 90% of people are right handed also

• Do you know any Physics teacher as good as you? 😊

• michel van biezen

• hey Nancy hope u doing well, actually I jst wanna knw why u love pi so much

• Great teacher: very consequencial,slowly speaking and clear. Thank you!

• …and pretty, to keep eyes focused on the front of the class…

• Completely different from mine

• How would you find x^3+3x-2? You did not cover what to do if there was a constant in it?

• when using the power rule, you do power times the number and -1 from the power. The constant -2 just turns to 0.

• Nancy, your videos are so helpful! They have really helped me through Calc1. But there’s something I must know…
How do you write backwards? It blows my mind every time

• She flips the video lol

• OMG Thank you so freakin much!!! This rely helped me.

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• I don’t know what you’re talking about but I could watch for hours. My math capability is: Duhhh!!!

• Thank u, I love u

• Thank you so much for this, it helped me a lot with my homework and hopefully, you have more videos as I get more into calculus.

• I just wanted to say thank you Nancy. I am not sure why my professor made this so unnecessarily hard. You are the Math Goddess.

• Thank you so much! It’s amazing how simple it all is but I was dying over here trying to figure this out

• Thank you very much !!!! Ma’am

• Forever I wish you were my calculus teacher!

• Can you please do a video of finding the slope of a curve without using derivatives. Using limits plz ;-;

• use either the alternate form or the definition of a derivative to find the derivative and go from there. The one with h in it

• are you really writing backwards

• lol the video is just flipped ..

• A tangent to straight line is the same straight line !

• so hot

• On the third example f(x)=(secx)^2 , shouldn’t f'(x)=2(secx)(secxtanx)^2 ??

• Can you do some linear algebra videos too? Please Mrs.Teacher 🙂

• Her name is Nancy

• Or you could write x/(x-1) as x.(x-1)^-1, if you forget the division formula a lot like me 🙂 Then use product and chain rule in order to solve. It is a different function but it works.

• Am I completely stupid or just exhausted from a very long day but…how the hell is she not writing backwards on the board?

• You are really beautiful and video is very helpful though.. thanks

• hey this was so useful!!! Can you do one with tangent lines in regards to implicit differentiation?

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• Thank you! This really helped with my hw!

• Thank you very much for the concept. The video was in detail and easy to understand.

• i got lost at the second question

• watch a vid on quotient rule

• is this the same as the equation of a tangent?

• I think u could be a little more active n interactive… But ur gr8..

• Soooo pretty 😍 thanks for the vid !

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• We learned how to do this before knowing what derivatives are, and holy shit it would take like 5-10 mins. This method takes like 1 min thank u so much

• Hold on I have a question. Was I supposed to look at the equations ? If yes, then I’m glad I didn’t waste 19 minutes by ignoring the equations and focusing somewhere else throughout the video. If u know what I mean.

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If you agreed click like

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• Madam you are great explain everyone step easily

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• By constructing a tangent line, we can calculate the slope of a curved line at a given point ? Explain this to me please? Oh and you can have all my pasta. You have an amazing sense of humor in the Q & A.

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Mam…!

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• what would you do if you dont have the y value given though?

• Solve for it

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• Thank you!

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• HI Nancy! so what would it be in this case.. Your answer would be greatly appreciated

x ̈ + x + αx3 = 0

α = 0, 2, −2

Regards

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• I always found point slope formula unnecessarily complicated. I find it much easier to use the well known y=mx+b substitute x and y with with your (x,y) ((1,3) in the first example) coordinate values, and substitute m (5) with your slope value and solve for b.

So 3=5(1)+b —> b=-2

Now rewrite y=mx+b with your b and m value so y=5x-2
ezpz

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OK this is the thing, to overcome your fears, you need to be confident and free yourself because people are watching you live, so you need to practices that and anyway thank you for the lesson God bless you madam.

• Dear Nancy, You’re beauty makes me strive to become a better math student. With every video, I fall more in love with you. Keep up the good work and keep making videos 😉

• maybe you also want to make a video for our professors about “how to teach math CLEARLY to your students”

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• I just want to see a 5 minute making of post showing us how you do the mirror writing trick.

• eitughtueir she probably has a mirror that u could see through both sides and then placed her camera on one side and writes on the other just my guess

• really in depth, but it was easy to follow and understand. In class, I was lost for a week when learning this topic, but after I watched this video, everything just clicked

thanks! 🙂

• this is probably just me being a bit dumb but are you writing backwards?? i imagine theres a glass plane infront of you and for the camera to pic it up in the correct sense?? if thats the case what a badass ahaha,, thanks for the helpful videos :~)

• OHHH i forgot editing software exists wow its just flipped around

• Happy to say, your amazing Nancy Pi !

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• I think it would be easier to represent sec(x) as cos(x)^-1 . So we have like f(x)=cos(x)^-2. And obviously we get the same slope 🙂

• thank you so much

• Finally a good one.

yeah, I mean the chick

• Get lost idiot

• Would it be wrong on the second problem to cancel x-1 on top and one of them on the bottom to get
-x/x-1?

• Hi. I have a question. Is the equation wrote on the links section written correctly (x-x1=m(y-y1))? Thank you so much.

• Thanks mam,
I like yr way to teach.

• i love to watch your videos. not for the knowledge but for you only. you are very beautiful.

• For the genius people who keep sayin ohh how can she write from right to left… actually she is writing from left to right the secret is on the camera she is reflecting her video so it seems like she is writing from right to left.

• how you can do that ? writing from z to a ?

• thank you for the videos!

• Nancy, you’re everything I’d want my daughter to be!!!! You’re the best!!!

• can u help me with Equation of secent line please

• Basically, the secant line approaches the tangent as the difference in x values approaches 0. I explain more at

• Speaking slowly nice

• If only x is given and y is unknown, what shall we do then? Here (1,3) is given

• I want to kiss you.

• Can u pls use a darker color….pink is not easy to read ..thx!..
Thank you for sharing your beauty and brains ….you make learning a joy !

• THANK YOU

• You need to start wearing glasses. even if you dont need a prescription, you can pick some non scripts up at Walgreens. Get some cute ones.

• why?

• How would you graph that after finding the answer, do you have a video for that? Thank you, Nancy!

• Basically, graph the parent function, and then use the points associated with the tangent line to graph the tangent. I explain more at

• Luv❤️U!

• I’m trying my best to learn calculus with my last remaining brain cells. Thank you Nancy for this very comprehensive video.

• Hey!! Very very thanks for helping me for my question which will help me to crack CBSE board exams.

• Good

• thank you. explained it perfectly in a way that I can understand easily.

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• my teacher overcomplicated thee FUCK out of this, it’s so simple.

• Thank you so much!

• wish you could do a more complicated problem.

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• she is hot and beautiful

• The basic Example really helped me get underway on my derivatives homework. I’ve never had to learn them until Uni, and still didn’t grasp it in the week I had lectures about it. Thanks for an awesome video.

• so how do I find the equation of the tangent line:

f(x)=7e^−0.4x  at x=7?

• allah razı olsun şu güzelliğe bak

• mam plx explain on the logorithm ALSO HOW TO FIND THE TANGENT LINE WITH THE HELP OF DERIVATIVE FORMULAS .

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• great…

• At first i was confused how she was writing backwards but she must flip it in post lol

• Very helpful thank you so much 😊

• Thank You Nancy

• If my maths teaches was pretty like you.. I’ll get an A for sure 😀

• she teaches us happily

• You won’t get a clearer explanation than Nancy P’s demo! Well done !

• Thanks mam please video on the topic maxima and minima please💐💐💐💐💐💐💐💐

• Thanks nancy

• Mire ha mi eso es suma por motivo de un forma de hoyente

• is she writing backwards?

• Better than my actual college profesor

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• thank you so much!! you’re my favorite youtube math tutor now!! please don’t stop using you’re platform to share and help the students like me who’s having a hard time understanding the lessons that are thought by our actual teachers. God bless you! xoxo

• Right! Why is it hard for math teachers to explain things, but people on youtube can explain it so perfectly?!?!? I’m in a hybird calculus course and the online learning lessons is just screenshots from a textbook and doesn’t actually explain a lot of steps and why something was done. Then there’s examples of a lesson that was never taught how to solve. It’s just assumed I’m guessing. Then you have to pay another online website to complete class assignments……
Capitalistic america and lazy professors!

• LOCD&LOOSE ugh, ikr 🙁

• I don’t under the last part when you put 5x – 5? Where did the – 5 come from Nancy

• Graceful.

• how can you write in the opposite direction or is it some software which is doing this?

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• This was extremely helpful. I didn’t even need to watch the full video. Thank you for explaining it thoroughly!

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• does she have to write backwards for it to not look flipped for the camera?

• Visual ASMR.

• How does she draw everything backwards

• I’m certain she flipped the video. I was wondering the same thing!

• ily ty

• Thanks a lot, NancyPi. As always, you make Calculus easy to understand.

• you’re pretty……………….smart! haha

• علي الصديق الاسطوره

• If you were my teacher, wow, there’s no way I’d be able to concentrate on math. I know you must get that all the time. Have you ever noticed that you’re actually a very beautiful woman? Sorry! LOL

• Suddenly, I am this much more interested in math.

• She’s hot no cap now I wish she was my calc teacher

• god bless you for not littering your board with d/dx’s and stuff.. like obviously we know whats going on you dont need to make it look so messy

• This was extremely helpful! Thank you!

• Thanks that simplified a lot!

• is she writing backwards??? how is this video possible??? im so confused

• Thanks

• a linear…..line 🤣😂😅. Great video. I’m learning so much calculus. I don’t even know why. I’m not even in school!

• I swear, I you were my professor, I would fail your class! ‘Cause I wouldn’t be able to keep my eyes off you!

• I love how she speaks slowly and gives multiple examples. It’s almost like she knows I’m a moron and yet she does not judge me.

• A true teacher. 😂

• shes judging u silently

• HAHAHHAHA hilarious comment

• I feel you bro

• I really appreciate how you explain each step and include the simplest of things like labeling the slope as ‘m’ because some of us get confused when only slope is said.

• There is a special place in heaven for you madam – many thanks!

• you say the third step is to find the y coordinate yet because you decided to already include it in this example you just skip over it. Would have been nice to see that step in case a problem doesn’t give the whole point.

• Really! Thank you!

• First you use the left hand, write in the opposite site and you do calculus so you are way to smart.

• Thank you for doing these. You are an awesome teacher. I wish I had these videos when I was taking math classes in college.

• you lost me at “find”

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• Can you make a video specifically on subtangent

• Thanks for the explanation

• When you mistake “at” for “a+”.

• Thank you queen

• saved thank god–

• Hi mam

• Hi guys, am Nancy

• She does too much for her to just ask for a like OR to subscribe. For her work she should always say to like AND subscribe. Love the material she puts out!

• I find it difficult to see when u use pink colors……

• 10:13, f(2) = 1

• Madam, all your lectures are so lucid! We did not have calculus in high school, and as a clinician I needed to understand the dynamics of the on-going Covid-19 pandemic. I have been able to learn calculus from your lectures, and have been successful in understanding the epidemic curves much better. thanks so much!

• Brain: look at the way she solving
Heart: look at her beauty

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• Hi

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• “it’s a linear line”

• So so helpful! Thank you!

• prof has a great saying for quotiant rule, ” low dhigh less high d low, down below, low square we’ll go.

• i lost

• How does she write backwards on the board so perfectly?

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• Did anyone else notice see only brought square root 3 over and not sqr root 3/2. Or did I miss something?

• Disregard my previous comment. I had a slight senior moment.

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• Dude, she wrote backwards the whole time.

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• These are very helpful, just a note, pink is very hard to read.

• Hi ma’am
I must say commendable video.
Love from India 💖
By the way,Which board are you using. Where can we buy one.

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• 0:24 It’s a linear…..line

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• beautiful, simple, to the point. Thanks

• Very good teacher! Thank you nancy!

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