Find the Equation of a Tangent Line Using the Definition of a Derivative

In this example, we find the equation of a tangent line by using the definition of a derivative (that is, we find the derivative the LONG WAY)!
Of course one could find the derivative using the shortcut methods but in the beginning of a calculus class, it is often required to use the definition. Of course, if you know the shortcuts, use those to verify your solution!
Note that the derivative is used to find the slope of the tangent line. After that we are simply using point-slope form of a line.

[ColdFuse]

0:55 You can actually skip this entire section of finding f(3)' by using the power rule:

*d/dx (X^n) = nX^x-1*

so:

f'(X) = X^2 - 3:

- X^2 = 2X^2-1 = 2X^1 = 2X

- 3 = 0 (<-- the derivative of any constant (meaning anything that doesn't have an X attached to it) is always zero)

so
- f'(X) = 2X

[abdullateef]

PatrickJMT, you are really a legend man! You are really helping me with my 2nd midterm for calc 1 this semester that is based on derivatives. I would like to thank you for your amazing effort to post these videos on YouTube for free to help students like me! Please keep the videos coming your work is greatly appreciated :)

[xmwmjsg]

Can't thank you enough. Somehow this is better than anything else I've watched. Thanks man.

[TonycoolsChannel]

Thank you, this video clears things up for me :)

[plumstreetmusic]

The tangential slope is simply 2x = 2x^1. I would would explain visually it as "copy" the exponent and bring it around front (a scalar). Then subtract 1 from the exponent.

[khaleelal-ashhab]

a more simple way
the gradient of the line can be found by substituting 3(x) in dy/dx
u then know that the gradient is equal to 6
so y=6x+c
u can find c by substituting x and y given in the question

[XenosFFBE]

Instead of doing all that limit stuff to find the derivative, can't you just do it in your head like usual? Or is doing the limit equation normal?

[mathbeginners]

This actually what i am doing now. This video will keep me reminder.

[ldcartman]

Just in time! You're a life saver!

[javeriaanis]

Can we use any other method other than derivative to find the equation of a line tangent to a curve and parallel to another line?

[milesspearman]

Your videos are awesome!!! Is there a video explaining why you did 2x+0 at the end, right before f(x)

[magznificent]

how do you find the equation of the tangent line without using derivatives but instead using the algebraic definition of tangency.

[HomeDepotDade]

theres a much simpler way of doing this. Just do the derivative of your original function, Plug in x on the new derivative equation and that equals your M. Then just plug in x1, y1, and M and thats all.

[raheem]

In London we are taught a different way which is makes more sense

[PixelZackOnHabbo]

Hey man, thanks for making this video. I dont know if u like compliment but thank you so much :)

[DanielJimenez-pznp]

I completely lost him at 1:17 when he just so casually said "We're gonna replace all the X's with (x+h) so we'd have (x + h) ... Squared-"

Me: *WOAH* What is he doing and why

He proceeds to do a bunch of arithmetic with the variables instead of plugging in. Now I'm lost and back to square 1 on not being able to figure out the problem... The next step which he finishes at 1:36 I am just drawing a complete blank on why he did that

[erikw]

Why wouldn’t you just use the power rule?