Derivative symbolab

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Computing an integral, symbolab showed the following derivation:
y\left(2-\frac{y^2}{2}\right)=y\cdot \:2\left(1-\frac{\frac{y^2}{2}}{2}\right)
I was unsure how the RHS of the equality was computed. I'm aware that 2 was taken out of the parenthesis, but I was unsure of the jump from \frac{y^2}{2} to \frac{\frac{y^2}{2}}{2}.
Any hints?

Hello, first of all, sorry foy my bad English.
I was studying implicit derivative and everything went great until i found this problem: x^2= x-y/x+y
I thought that is was easier if I rewrite the expression as x^2(x+y)=x-y, so the final expression would be x^3 +yx^2=x-y. I did it and everything was ok until i look at the answer that is in the book and its completely different, then i went to Symbolab and in fact the answer that symbolab gave me from that problem was the same that i got, but when i put x^2=x-y/x+y in symbolab it gave me another answer that is the same that it is in the book. So i don't kno whay this happened, I believe that x^2(x+y)=x-y should be the same as x^2=x-y/x+y, but when i implicit derivative both expression it led me to a different answer. Is there any rule in this type of problem that i don't know?
Thank you!
(Sorry for any grammatical mistake)

I have to find H'(x), where H(x)=2xsqrt{x}
The solution is 3x (written in my book)
I tried to solve it and got the right solution.
But when I plugged in the problem on “symbolab” and compared the way I solved it to the way symbolab did, it seems like I did it wrong.
Symbolab: So initially it simplified the expression first before finding the derivative. It took the constant 2 out. And found the derivative of x^{{\frac{2}{3}}} - much simpler than what I did
The way I solved it was using the product rule, because I thought 2x could be seen as a function and \sqrt{x} could be seen as a function.
How can I learn from this next time - to know which kind of method to use?

I have an equation in front of me of the format Y"-4Y-12Y = sin2x
The given solution (by symbolab) Y(X) = c1e^-2x +c2e^6x -(e^5x)/4 + (e^5)/12.
I was able to get the first two terms of the solution perfect but the non-homogenous particular solution is giving me trouble. So far I've looked at the table of likely particular solutions and decided that since my non homogeneous term is sin2x, my guessed solution would be: Acosbx + Bsinbx.
I took the first and second derivative of this guess and substituted it into my initial equation to get to this point
-13Acosbx -13Bsinbx +4Asinbx +4Bcosbx = sin2x
From here I'm totally stumped. I have no idea what I'm doing. I don't know I've what time be done up to here is good. Please assist. I am in Australia so my level of calc is different to what the terms in America are but I've reached the point of modelling problems with differential and doing double integrals.
Thank you for any help you can provide.

So function is x on the power of x on the power of x
y = x^x^x .
So I took ln (natural log) on both sides and I got
lny = x^2lnx
I took derivative of lny which is x(2lnx + x)
In general cases ln'y(x) (derivative of lny(x)) is equal to y'(x) / y(x) so y' must be equal
y' = ln'(y) * y
I my case I got y' = x^x^x . x(2lnx + 1)
​
But I checked on Symbolab and it gave me x^{x^x}\left(x^x\ln \left(x\right)\left(\ln \left(x\right)+1\right)+x^{x-1}\right)
I'm pretty sure they're different results because seconds answer cannot be simplyfied anymore.
On symblab software use this rule:
a^b = e^blna
What am I doing wrong?
Edit: second answer didn't come out as I expected so here's the link: https://www.symbolab.com/solvestep-by-step/%5Cfrac%7Bd%7D%7Bdx%7D%5Cleft(x%5E%7Bx%5Ex%7D%5Cright)?or=input?or=input)

This is pretty simple, but I keep getting different answers. It’s finding the derivative of f(x^2). I think it’d be 2xf’(x^2), but I also saw online that it’d become 2f(x) from Symbolab somehow. I assume you’d use Power Rule and Chain Rule, but maybe I’ve done it wrong. Which is it? Or is it different?

Hi all,
I'm taking the Calculus 1 course at Sophia with the intention of transferring it into WGU.
For the purpose of graduating from WGU, how important would you say it is for me to have a firm grasp on every way of solving a derivative, versus simply knowing how to use a CAS to achieve the answer?
I'm at a point in the course where I can either start memorizing all of the techniques for solving a derivative, or I could simply advance through this unit by using Symbolab to calculate the challenge+exam answers.
Conceptually I think I have a good understanding of the math, I’m just trying to decide if it’s going to be useful to memorize these various techniques for solving every problem on paper.
tl;dr: will I be setting myself up to struggle in WGU's remaining CompSci math courses if I keep reaching for a calculator to solve a derivative?

Anyways, here are the two problems: 1. What is the first derivative of ln(ln(y)) + ln(y) = ln(x)? A. 2y/(x+y) B. y/(x-y) C. 2y/(x-y) D. y/(x+y)
I tried differentiating it and got y*ln(y)/(x*(1+ln(y))), which is close to choice D. I did it by first simplifying the ln using ln(xy) = ln(x) + ln(y) and then using the product rule. It's an implicit function, therefore, I had to isolate y' after differentiating them, which led me to what I have right now. The problem is, I can't find any way to remove ln to get to fit choice D. I even tried to evaluate each selection using Symbolab by letting it solve the problem initially but got everything wrong. Not a single one of those is correct according to Symbolab. And the answer I got is the same as the one that showed up using Symbolab. Therefore, how?!

1. Differentiate y = 2e^x*8^x A. 6.16e^x*8^x B. 2e^x*8^x C. 3e^x*8^x D. 4.16e^x*8^x

I observed that I can simplify the given expression to (2^1+3x)(e^x) and I differentiated from there but ended up lost. I tried to study the solution I got from calculators but every single of them doesn't give the same result as the choices. I also tried evaluating each one of them and found out A was the real answer. But, the thing is, I have no idea where that decimal came from. Where do I even start?

So I encountered the function f(x) = xx . I wanted to find its derivative at the point x = 0, which is ineed continous.
If I take x = sqrt(x^2) the I get that the derivative of f(x) is f´(x) = (2x^2)/x. We are dividing by x so x cannot = 0.
However, if I take x = x^2, x => 0 and -(x^2), x < 0 . And I derivate each part I get f'(x) = 2x, x => 0 and -2x, x < 0. So, by taking the definition of x I get that x can = 0, where the derivative is 0.
Desmos, Photomath and Symbolab say that no derivative at x = 0. My university says there is, Wolfram Alpha says the derivate is 2x implying that there is. I have no clue what is the real answer now, could anyone tell what the *real* derivative is and why to each source differ.

I know many people use Straighterline so I wanted to post a study.com review. I just finished all the quizzes and tbh found it relatively easy. I watched the Math Sorcerer Calc1 over 3-4 days which covered almost all of it.
I didn’t really do the problems but followed along in my head trying to guess the next step out loud as he went over example problems. I had a good conceptual grasp of what was going on from that and enjoyed having an overview before tackling problems.
I didn’t use a calculator for the course except for some natural logs but even then, iirc all answers are left in pi, e, or log form, so you don’t need to compute the actual decimal number.
The entire point of the course is really integration. Finding anti derivatives and derivatives. You learn it piece by piece and put it all together at the end.
I never really referred to the SDC material. I sometimes Glanced over it but if I had an issue I simply searched YouTube for the method and watched the organic chemistry tutor who usually popped up as a first result.
By going over the math sorcerer series first I pretty much had a good idea of what to do right away, and needed a 5-10 minute video from the organic chemistry tutor occasionally for more explanation.
For example, U substitution took me a minute to get bc I was just overlooking a simple detail for a while.
Trig substitution took a day of going wtf bc I was tired, and need to just go back over trig identities.
Finally, Symbolab $5/month i would suggest.
If I was stuck I would plug in the anti derivative formula and follow it step by step. Then tackle the problem to ensure I retained it.
Overall the hardest part is making silly algebra mistakes, forgetting you pulled out a 1/2 constant for example.
If you’re worried about calc 1 I just wanted to share this.

Question y=1+2x/sqrt(1-x^2)
where u = 1+2x
where v = sqrt(1-x^2)
where v'=-(x/sqrt(1-x^2))
did the quoteint rule and reached a conclusion of 2+x/sqrt(1-x^2)^2

Question y=1+2x/sqrt(1-x^2)
where u = 1+2x
where v = sqrt(1-x^2)
where v'=-(x/sqrt1-x^2)
did the quoteint rule and reached a conclusion of 2+x/sqrt(1-x^2)^2

Hi Guys, i am learning calculus and I find myself using both Wolfram Alpha and Symbolab for solving math questions (when i can not do it by myself or when i want to check if i solved right) I am thinking in paying for a subscription to use the premium features of one of those two softwares, basically I would be using it for calculus ( Precalc, Calc I,II and III).
​
¿Which one would be better for me? (Better as in more capable of solving limits, integrals and derivatives )
​
Thank you in advance.

Hi Guys, i am learning calculus and I find myself using both Wolfram Alpha and Symbolab for solving math questions (when i can not do it by myself or when i want to check if i solved right) I am thinking in paying for a subscription to use the premium features of one of those two softwares, basically I would be using it for calculus ( Precalc, Calc I,II and III).
¿Which one would be better for me? (Better as in more capable of solving limits, integrals and derivatives )
Thank you in advance.

Just wanted to share some tips for incoming 1st year students planning to take the harder proof-based math courses like MAT137, 157, 240, and 247:
1. Preparation is worth it
Many people obviously struggle with proofs at the start, but you can alleviate much of that pain by learning basic proof techniques, and reading a bunch of worked examples of proofs. Nobody is born with the faculty to do proofs (well... maybe almost nobody). And while the courses MIGHT go over some basic proof techniques at the start, the instructors won't hesitate to get straight to the course content after a week or two. Learning the building blocks of writing proofs will make it a lot easier for you to learn the content.
Two good resources to learn proofs are Velleman's How To Prove It or Hammack's Book of Proof. It is important to do problems of course, but it's even more important to study a ton of examples of basic proofs using different techniques. With enough knowledge, you should be able to recognize common themes like:

• if a theorem involves the natural numbers, then you should try to prove it using induction,
• if a question asks you to prove something exists, you should most likely try to construct an example, or
• if you are having trouble even starting to prove something directly, you should consider a proof by contradiction

and so on. If you don't know what any of those mean, you can find the secrets in the books I've mentioned. But knowing these themes will make solving problems way easier.
2. Pay attention at the start!
Even if you already know how to do proofs, every field of math has its own basic techniques that you want to get used to sooner rather than later. The theorems that you'll learn at the start might be trivial statements, even boring. But PAY ATTENTION to their proofs and let the techniques sink in. Later on, you'll see the the same techniques used to prove basic stuff in proofs of harder theorems.
Furthermore, don't immediately look to do the hardest problem you can find in the textbook. Do a lot of simple problems that use those basic techniques. It is difficult to progress to the harder content without a solid grounding in the basics, even if you think you're a math wizard. You can only trip on the banana peel if you don't pick it up beforehand.
3. Do problems... but study a lot as well
Paul Halmos said "The only way to learn mathematics is to do mathematics." And while there is a lot of truth in that quote, it is also at the same time an exaggeration. Solving a lot of problems reinforces your knowledge, and allows you to solve similar problems in the future more quickly. But it is equally important that you don't skip the studying. You want to have the prerequisite knowledge needed to tackle those problems.
Say for example, you are trying to prove something involving limits. It is perfectly fine if you struggle with the problem – math is hard! But if you're staring at the sheet of paper and have no ideas because:

• you don't even know what a limit is, or
• you only know the definition and have not learned the ways to manipulate limits,

go back to your lecture notes and thoroughly learn the content again. This ties back to the last point – KNOW the basic techniques of the field, so you don't have to reinvent the wheel when all you really need to do is fix a tire.
4. Computation is important
If you hated plugging and chugging in high school, you'll probably like proof-based courses much better. But that doesn't mean – even in proof-based courses – that learning how to compute and manipulate things isn't important. You want to be comfortable at elementary algebra before you even enter the first lecture. You don't want to see the instructor add something to both sides of an equation and think, "what the fuck?". It should be second nature.
And as the course progresses, you must be able to perform the basic calculations quickly. Don't be the one who can prove the Heine-Borel theorem in the shower but who can't even do a basic derivative or integral without Symbolab. There WILL be computation questions in the midterms and exams, and while there are beautiful one or two-line proofs, those are comparatively rare – most proofs will require at least a few gnarly computations or manipulations.
5. Prepare now, or prepare to work hard
Proof-based courses are scary for a lot of students. But consider why they are scary in the first place. Because proofs are unfamiliar! What can you do to deal with that? You familiarize yourself with proofs. And while I could've just said "it'll be fine, you'll get the hang of it", the truth is that – unless you're a genius – it will be a struggle if you don't prepare beforehand or at least prepare yourself to work a lot at the start of the course, especially if you're going to do the specialist courses (157/240/247).
6. And finally a very practical tip...
Use lots and lots of paper. Really. You don't want to have 10 different things in your working memory when you're solving a problem.

I've tried everything i have to prove that:
(Sin(3x)/Sin(x))-(Cos(3x)/Cos(x)) = 2
WolframAlpha says is true, as well as Symbolab and Mathway, but i need help going step by step, i cant use integrals nor derivates, i know theres some way using just trigonometric stuff, any help would be greatly apreciatted, thanks beforehand

Hello, I have two different functions here: https://imgur.com/a/7wxOTup
When asked for the derivative, my answers for both problems aren't accepted. They are straight from symbolab. Am I missing a step? This is the first time my derivatives haven't been accepted.
Thank you for your time

so according to my furmolas page the derivative of e is (e^x)'=e^x and nothing more but according to things i did in symbolab its actually (e^ax)'=a*e^ax as well, is it true? im not sure myself, thanks in advance

Hi all,
I am trying to bump up my math skills step by step and now I have to exclude as many factors as possible from parentheses.
The question is factoring 2(a+3)+4(a+3), which I derive by (what I learned earlier) calculating each combination 2a + 6 + 4a + 12 6a + 18 6(a+3) which is the correct answer.
However, the explanation found by Symbolab does the following: 2(a+3)+4(a+3) 2(a+3)(1+2) 2(a+3)(3) 6(a+3). But no explanation about this transformation of 2(a+3)+4(a+3) 2(a+3)(1+2) is given (even with my paid pro account that should most often explain all the steps). What are the steps or rules that they use here?
Thanks in advance! :)

hey everyone, I've asked this already on stackexchange but I don't quite understand the logic behind the response I've been given. I'll reiterate the question and explain my attempts, here's the original post - https://math.stackexchange.com/questions/4322716/finding-the-derivative-of-an-integral-of-a-continous-function?noredirect=1#comment9010893_4322716
In the question I've been given a function:
g(t) = (7 + sin(t)/t for every t !=0, 8 for t=0)
and I've been tasked with calculating the derivative of F(x) which has been defined as:
F(x) = ∫ g(t) d(t) between 0 and x^2 -6x.
My work:
I've noticed that g(t) is continuous since sin(t)/t will approach 1 as t approaches 0.
In my first attempt I've tried using newton-liebnitz:
F(x) = g(x^2−6x)−g(0) = sin(x^2−6x)/(x^2−6x)−1, I've derived and submitted this and got a wrong result. (I verified my derivative with symbolab).
a person there replied I was using it wrong since "You are misusing the Fundamental theorem of calculus. F(x) is not g(x^2−6x)−g(0); this is true if you use a capital G, for G a valid indefinite integral/antiderivative of g. But what about F′? " - so if I understood this correctly, F isn't really the antiderivative of g I am not sure I understand why but maybe because g is comrpised of two different functions? I'd love a clarification. Bottom line is that I can't use newton liebnitz.
I wanted to go into the limit definition of an integral but that was really complicated and I felt like there must've been an easier way.
I've tried using this formula:d/d(x) ∫f(t)dt (from u(x) to v(x)) =v′(x)∗f(v(x))−u′(x)∗f(u(x)) and got this :
F′(x)=g(x^2−6x)*(d/dx(x^2−6x)).
to me it seemed fine but I seemed to have got it wrong (I've got three attempts in total, used 1 for newton liebnitz and another one for a mistype - I can't verify), according to a commenter the result should be-
F′(x)=g(x)*(d/dx(x2−6x)), which he got by using the fundemental theorem (does he refer to Newton Liebnitz? same as me?) and by using the chain rule - (which I believe is the one I've written above, right? )
I've asked for clarification and waited a day but got no response.
tldr - I don't understand the logic behind the proposed solution, why my second solution ended up wrong, and if my understanding of the first solution's error is complete?
I know you don't appreciate it when people send solutions, but is it possible for someone here to confirm the validity of my own solution post recieving hints? - this hw is graded and I've used all my attempts but one.
or does anyone know of any calculator that takes in these types of integrals and spits out a result?
this isn't a test nor an exam, I've shown my work. I hope this is enough.

Hello,
I have computed this derivative and I come up with the answer of sin²(x)-cos²(x).
I like to check my answers with different online derivative calculators, and each website has given me a different answer.
The answers I have gotten from website calculators are:
Wolfram Alpha: -cos(2x) OR 1-2cos²(x) OR 2sin²(x)-1
Symbolab: -cos(2x)
TI-Nspire calculator: 1-2cos²(x)
Derivative-Caclulator.net: sin²(x)-cos²(x)
​
So are all these answers equivalent in some way? Am I missing something here?
This is how I worked it, maybe someone can point out where I am right/wrong.
d/dx[-cos(x)sin(x)+C] => d/dx(-cos(x))sin(x) + (-cos(x))d/dx(sin(x)) + d/dx(C)
=> sin(x)sin(x) + (-cos(x))cos(x) + 0 => sin²(x)-cos²(x).