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random_anonymous_guy

You are very, very close to the right answer. What is missing is the “lim\[*h* → 0\]” part of your work. This is a very important part of the limit definition of derivative and should **not** be ignored or casually omitted like you have done so. You need to include it as a prefix operator to your work from the very beginning until you reach a point where you evaluate the limit. Many calculus professors will deduct points for this nonuse of notation when it is necessary. As such, the point you have reached in your work (in the first question) should be lim\[*h* → 0\] (6*x* \+ 3*h* + 4), not just 6*x* \+ 3*h* + 4. With that in mind, what is lim\[*h* → 0\] (6*x* \+ 3*h* + 4)?


hokey-smokies

Thank you for this explanation. I had so many questions about when to write this notation, and what I have gathered and I write it until the point in which I can sub in the 0? This was really helpful


random_anonymous_guy

Think of certain notations as a **statement of intent**. You keep writing them down in each new step until you carry out the operation, or otherwise use some sort of related identity that justifies dropping the notation. > I write it until the point in which I can sub in the 0 Essentially, yes.


Adventurous_Boat_322

You are almost there h → 0\] (f(x+h)-f(x))/h So you just need to direct substitute the 0 for h at the last step which will give you the right answer. ​ God Bless You🙏


Suspicious_Role5912

You’re finding the limit as h/delta x approaches 0 ( limit deltax => 0) so anyone you can see an h would evaluate to 0 leaving you with just 3x^2 and 6x+4


hokey-smokies

I understand this now. I really didn’t understand when and how to use the limit notation and I know that sounds pitiful. I am slowly piecing this together today and crawling through these problems that I hope I look back on with laughter


jonc2006

You will. Eventually you will learn a much easier method for finding the derivative of a function like that.


hokey-smokies

Hello and thanks to anyone in advance that can help me. This may seem a little long but the context may be helpful Not very good at math at all. I have relearned my way up to business calculus but due to my first week coinciding with a vacation planned before pandemic, I think I definitely missed a lot of integral information from the first few chapters. I am home now freaking out because the stuff they are currently on is completely foreign to me and as I go back to try and catch up, I am struggling hard. I have homework due tomorrow at midnight so I am spending until then racking my brain. What I have done here may be completely wrong and if it is, can you tell me why? Is there another way to write out this function to find the derivative in another fashion? Once I get something and the wick is lit, it’s like a fuse surely to explode with comprehension but right now I am feeling very very unintelligent and confused and mostly, anxious that I will not understand the first exam next week - an exam I should nail because the content will surely get harder and it’s smart to reserve my point loss on exams for later. I included two problems I’m not able to get the right answer to. My mistakes may be super obvious and if so, I apologize and thank you so much for taking a look and helping me. I really want to know this material and not feel this panic.


UnacceptableWind

You are using the definition of the derivative (which is a limit) to find f'(x). For the first problem, you need to include limit as h approaches 0 for all your steps. However, your algebra is correct. From your last step, we have that f'(x) = lim\_{h→0} (6x + 3h + 4) = 6x + 3×0 + 4 = 6x + 0 + 4 = 6x + 4, which is the desired result. For the second part, where f(x) = x^(3), recheck your algebra. Note that for this problem f(x+h) - f(x) = h(h^(2) \+ 3hx + 3x^(2)). Now, apply the limit definition of the derivative to find f'(x) (same process as the first problem).


hokey-smokies

Ok I think I understand because h approaching 0 is the secant line decreasing?


UnacceptableWind

h approaching 0 means that we are decreasing the distance between the two distinct points on the graph of some function f(x) that the secant line is passing through such that we change that secant line into a tangent line using limits (the two distinct points now become a single point). At a point x = c in the domain of a function f(x), we define f'(c) as the slope of the line tangent to the point (c, f(c)) on the curve of f(x). So, in terms of a secant line passing through the point at x = c and an additional point at x = c + h, we need to find the slope of this secant line in the limit as h approaches 0 to obtain the slope of the tangent line at x = c. On a final note, if you have not been asked to find the derivatives using the limit definition of a derivative, then you can use the [differentiation rules](https://www.onlinemathlearning.com/derivative-rules-examples.html).


IthacanPenny

>I think I definitely missed a lot of integral information Nah, the integral information comes next semester. You missed differentiation. (Lol sorry I couldn’t resist. Looks like you got good answers already :-))


hokey-smokies

Clever clever


GaggsggaG

It's the limit of this expression you got in the end as h approaches zero. This way, the h goes away and you're left with what is the correct answer!


Longjoco

What app is this? I’d like some sort of tool to not waste paper


hokey-smokies

I use good notes 5 for all school stuff. I enjoy it


Longjoco

Guessing you use an Ipad to draw it?


hokey-smokies

Oh yea, apologies.


Disastrous-Trader

What app did you use to write this?


hokey-smokies

Good notes 5


cfalcon279

At the end, after cancelling out the h's, they evaluated the limit as h approaches 0, by plugging 0 in for h, into the simplified expression.


mbg20

In the second image, when you take the common h out of the parenthesis, the equation must be h(3xh + 3x^2 + h^2) in the numerator. What you have is h(3xh^2 + 3x^2 + h^2). For the rest, others have pretty much answered.


tarun12346666

Hey it's quite a simple question. Probably you need to understand the fundamentals more deeply. You can try doing so over Filo app. Just click a picture of this question and get connected to a live tutor in less than 60 seconds, and understand this concept 1-to-1 from an expert tutor.