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Questions tagged [derivatives]

Questions on the evaluation of derivatives or problems involving derivatives (for example, use of the mean value theorem).

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How discontinuous can a derivative be?

There is a well-known result in elementary analysis due to Darboux which says if $f$ is a differentiable function then $f'$ satisfies the intermediate value property. To my knowledge, not many "...
189
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6answers
13k views

Why does this matrix give the derivative of a function?

I happened to stumble upon the following matrix: $$ A = \begin{bmatrix} a & 1 \\ 0 & a \end{bmatrix} $$ And after trying a bunch of different examples, I noticed the ...
158
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5answers
6k views

Why can't differentiability be generalized as nicely as continuity?

The question: Can we define differentiable functions between (some class of) sets, "without $\Bbb R$"* so that it Reduces to the traditional definition when desired? Has the same use in at least some ...
128
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6answers
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Derivative of sigmoid function $\sigma (x) = \frac{1}{1+e^{-x}}$

In my AI textbook there is this paragraph, without any explanation. The sigmoid function is defined as follows $$\sigma (x) = \frac{1}{1+e^{-x}}.$$ This function is easy to differentiate ...
124
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1answer
112k views

Derivative of Softmax loss function

I am trying to wrap my head around back-propagation in a neural network with a Softmax classifier, which uses the Softmax function: \begin{equation} p_j = \frac{e^{o_j}}{\sum_k e^{o_k}} \end{equation}...
122
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2answers
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Discontinuous derivative. [duplicate]

Could someone give an example of a ‘very’ discontinuous derivative? I myself can only come up with examples where the derivative is discontinuous at only one point. I am assuming the function is real-...
111
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6answers
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What did Alan Turing mean when he said he didn't fully understand dy/dx?

Alan Turing's notebook has recently been sold at an auction house in London. In it he says this: Written out: The Leibniz notation $\frac{\mathrm{d}y}{\mathrm{d}x}$ I find extremely difficult to ...
107
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9answers
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Proof that $C\exp(x)$ is the only set of functions for which $f(x) = f'(x)$

I was wondering the following. And I probably know the answer already: NO. Is there another number with similar properties as $e$? So that the derivative of $\exp(x)$ is the same as the function ...
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8answers
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Why is the derivative of a circle's area its perimeter (and similarly for spheres)?

When differentiated with respect to $r$, the derivative of $\pi r^2$ is $2 \pi r$, which is the circumference of a circle. Similarly, when the formula for a sphere's volume $\frac{4}{3} \pi r^3$ is ...
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7answers
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Does L'Hôpital's work the other way?

Hello fellows, As referred in Wikipedia (see the specified criteria there), L'Hôpital's rule says, $$ \lim_{x\to c}\frac{f(x)}{g(x)}=\lim_{x\to c}\frac{f'(x)}{g'(x)} $$ As $$ \lim_{x\to c}\...
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9answers
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What is the Jacobian matrix?

What is the Jacobian matrix? What are its applications? What is its physical and geometrical meaning? Can someone please explain with examples?
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3answers
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Construct a function which is continuous in $[1,5]$ but not differentiable at $2, 3, 4$

Construct a function which is continuous in $[1,5]$ but not differentiable at $2, 3, 4$. This question is just after the definition of differentiation and the theorem that if $f$ is finitely ...
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7answers
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$100$-th derivative of the function $f(x)=e^{x}\cos(x)$

I've got this task I'm not able to solve. So i need to find the 100-th derivative of $$f(x)=e^{x}\cos(x)$$ where $x=\pi$. I've tried using Leibniz's formula but it got me nowhere, induction doesn't ...
67
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1answer
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Thurston's 37th way of thinking about the derivative

In Thurston's superb essay On proof and progress in mathematics, he makes this observation: Of course there is always another subtlety to be gleaned, but I would like to at least think that I ...
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10answers
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Why not include as a requirement that all functions must be continuous to be differentiable?

Theorem: Suppose that $f : A \to \mathbb{R}$ where $A \subseteq \mathbb{R}$. If $f$ is differentiable at $x \in A$, then $f$ is continuous at $x$. This theorem is equivalent (by the contrapositive) ...
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6answers
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What exactly is the difference between a derivative and a total derivative?

I am not too grounded in differentiation but today, I was posed with a supposedly easy question $w = f(x,y) = x^2 + y^2$ where $x = r\sin\theta $ and $y = r\cos\theta$ requiring the solution to $\...
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6answers
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Is there a function whose antiderivative can be found but whose derivative cannot?

Does a function, $f(x)$, exist such that $\int f(x) dx $ can be found but $f' (x)$ cannot be found in terms of elementary functions. For example, if $f(x)=e^{x^2}$, then the derivative is easily ...
55
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1answer
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Elementary proof that the derivative of a real function is continuous somewhere

One can use the Baire category theorem to show that if $f:\mathbb{R} \to \mathbb{R}$ is differentiable, then $f'$ is continuous at some $c \in \mathbb{R}$. Is there an elementary proof of this fact? ...
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11answers
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Which of the numbers $1, 2^{1/2}, 3^{1/3}, 4^{1/4}, 5^{1/5}, 6^{1/6} , 7^{1/7}$ is largest, and how to find out without calculator?

$1, 2^{1/2}, 3^{1/3}, 4^{1/4}, 5^{1/5}, 6^{1/6} , 7^{1/7}$. I got this question in an Application of Derivatives test. I think log might be used here to compare the values, but even then the values ...
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When not to treat dy/dx as a fraction in single-variable calculus?

While I do know that $\frac{dy}{dx}$ isn't a fraction and shouldn't be treated as such, in many situations, doing things like multiplying both sides by $dx$ and integrating, cancelling terms, doing ...
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1answer
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The $100$th derivative of $(x^2 + 1)/(x^3 - x)$

I am reading a collection of problems by the Russian mathematician Vladimir Arnol'd, titled A Mathematical Trivium. I am taking a stab at this one: Calculate the $100$th derivative of the function $...
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15answers
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What are “instantaneous” rates of change, really?

Here's how I see it (please read the following if you can, because I address a lot of arguments people have already made): Let's take instantaneous speed, for example. If it's truly instantaneous, ...
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Why does the derivative of sine only work for radians?

I'm still struggling to understand why the derivative of sine only works for radians. I had always thought that radians and degrees were both arbitrary units of measurement, and just now I'm ...
47
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8answers
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When to Stop Using L'Hôpital's Rule

I don't understand something about L'Hôpital's rule. In this case: $$ \begin{align} & {{}\phantom{=}}\lim_{x\to0}\frac{e^x-1-x^2}{x^4+x^3+x^2} \\[8pt] & =\lim_{x\to0}\frac{(e^x-1-x^2)'}{(x^4+...
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6answers
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Why do all elementary functions have an elementary derivative?

Considering many elementary functions have an antiderivative which is not elementary, why does this type of thing not also happen in differential calculus?
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6answers
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A question about differentiating equations that are impossible to solve for a variable

I thought I had a good idea on why/how implicit differentiation works until I read the following passage in my Calculus book: Furthermore, implicit differentiation works just as easily for ...
45
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3answers
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Function that is the sum of all of its derivatives

I have just started learning about differential equations, as a result I started to think about this question but couldn't get anywhere. So I googled and wasn't able to find any particularly helpful ...
44
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10answers
8k views

How is the derivative truly, literally the “best linear approximation” near a point?

I've read many times that the derivative of a function $f(x)$ for a certain $x$ is the best linear approximation of the function for values near $x$. I always thought it was meant in a hand-waving ...
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3answers
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Is it possible for the derivative of a function to grow arbitrarily faster than the function itself?

We know that there exist some functions $f(x)$ such that their derivative $f'(x)$ is strictly greater than the function itself. for example the function $5^x$ has a derivative $5^x\ln(5)$ which is ...
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3answers
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Why, although these functions have the same derivative, do they not differ by a constant?

I calculated the derivative of $\arctan\left(\frac{1+x}{1-x}\right)$ to be $\frac{1}{1+x^2}$. This is the same as $(\arctan)'$. Why is there no $c$ that satisfies $\arctan\left(\frac{1+x}{1-x}\right) =...
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6answers
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If $f(x)=\frac{1}{x^2+x+1}$, how to find $f^{(36)} (0)$?

If $f(x)=\frac{1}{x^2+x+1}$, find $f^{(36)} (0)$. So far I have tried letting $a=x^2+x+1$ and then finding the first several derivatives to see if some terms would disappear because the third ...
42
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10answers
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Why do we require radians in calculus?

I think this is just something I've grown used to but can't remember any proof. When differentiating and integrating with trigonometric functions, we require angles to be taken in radians. Why does ...
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7answers
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What function can be differentiated twice, but not 3 times?

In complex analysis class professor said that in complex analysis if a function is differentiable once, it can be differentiated infinite number of times. In real analysis there are cases where a ...
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5answers
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Can a function have two derivatives?

I am a senior in high school so I know I am simply misunderstanding something but I don't know what, please have patience. I was tasked to find the derivative for the following function: $$ y = \...
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6answers
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Which derivatives are eventually periodic?

Which derivatives are eventually periodic? I have noticed that is $a_{n}=f^{(n)}(x)$, the sequence $a_{n}$ becomes eventually periodic for a multitude of $f(x)$. If $f(x)$ was a polynomial, and $\...
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7answers
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Function whose third derivative is itself.

I'm looking for a function $f$, whose third derivative is $f$ itself, while the first derivative isn't. Is there any such function? Which one(s)? If not, how can we prove that there is none? Notes: ...
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6answers
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Proving that $\lim\limits_{x\to\infty}f'(x) = 0$ when $\lim\limits_{x\to\infty}f(x)$ and $\lim\limits_{x\to\infty}f'(x)$ exist

I've been trying to solve the following problem: Suppose that $f$ and $f'$ are continuous functions on $\mathbb{R}$, and that $\displaystyle\lim_{x\to\infty}f(x)$ and $\displaystyle\lim_{x\to\...
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2answers
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Why aren't integration and differentiation inverses of each other?

Integration is supposed to be the inverse of differentiation, but the integral of the derivative is not equal to the derivative of the integral: $$\dfrac{\mathrm{d}}{\mathrm{d}x}\left(\int f(x)\...
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5answers
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How did Newton find derivative of basic functions before formulating systematic Calculus?

I think I read somewhere that Newton tried to find derivatives of basic functions like $x^2$ before formulating systematic calculus; how did/would he do it?
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15answers
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Why do differentiation rules work? What's the intuition behind them? (Not asking for proofs)

Differentiation rules have been bugging me ever since I took Basic Calculus. I thought I'd develop some intuitive understanding of them eventually, but so far all my other math courses (including ...
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8answers
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Does this pattern have anything to do with derivatives?

In 6th grade I was first introduced to the idea of a function in the form of tables. The input would be "n" and the output "$f_n$" would be some modification of the input. I remember finding a pattern ...
38
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1answer
912 views

If $\,\lim_{h\to 0}\frac{f(x+h)-f(x-h)}{2h}\,$ exists for every $x$, what does this imply for $f$?

Consider the function $$ f(x)=\left\{\begin{array}{rll} 1+x^2 & \text{if} & x \,\,\text{rational} \\ -x^2 & \text{if} & x \,\,\text{irrational}\end{array}\right. $$ Then, for $x=0$, ...
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2answers
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Differentiating an Inner Product

If $(V, \langle \cdot, \cdot \rangle)$ is a finite-dimensional inner product space and $f,g : \mathbb{R} \longrightarrow V$ are differentiable functions, a straightforward calculation with components ...
37
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1answer
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Can you take the derivative of a function at infinity?

Exactly the title: can you take the derivative of a function at infinity? I asked my maths teacher, and while she thought it was an original question, she didn't know the answer, and I couldn't find ...
37
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4answers
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Difference between gradient and Jacobian

Could anyone explain in simple words (and maybe with an example) what the difference between the gradient and the Jacobian is? The gradient is a vector with the partial derivatives, right?
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6answers
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Geometric interpretation of mixed partial derivatives?

I'm looking for a geometric interpretation of this theorem: My book doesn't give any kind of explanation of it. Again, I'm not looking for a proof - I'm looking for a geometric interpretation. ...
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4answers
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Is it necessary that every function is a derivative of some function?

I thought about this a lot and consulted a lot of people but everyone had contradicting answers. I am a high school student. please help.
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Notation of the second derivative - Where does the d go?

In school I was taught that we use $\frac{du}{dx}$ as a notation for the first derivative of a function $u(x)$. I was also told that we could use the $d$ just like any variable. After some time we ...
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5answers
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Why does the fundamental theorem of calculus work?

I've known for some time that one of the fundamental theorems of calculus states: $$ \int_{a}^{b}\ f'(x){\mathrm{d} x} = f(b)-f(a) $$ Despite using this formula, I've yet to see a proof or even a ...
34
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6answers
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What is the derivative of: $f(x)=x^{2x^{3x^{4x^{5x^{6x^{7x^{.{^{.^{.}}}}}}}}}}$?

I happened to ponder about the differentiation of the following function: $$f(x)=x^{2x^{3x^{4x^{5x^{6x^{7x^{.{^{.^{.}}}}}}}}}}$$ Now, while I do know how to manipulate power towers to a certain extent,...