# Questions tagged [derivatives]

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

20,849 questions
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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 "...
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### 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 ...
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### 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 ...
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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 ...
1answer
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### 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}...
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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-...
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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 ...
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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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### 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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### 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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### 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 ...
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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 ...
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### 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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### 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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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 $\arctan\left(\frac{1+x}{1-x}\right) =... 6answers 6k views ### If$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 ... 10answers 27k views ### 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 ... 7answers 15k views ### 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 ... 5answers 8k views ### 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: $$y = \... 6answers 4k views ### Which derivatives are eventually periodic? Which derivatives are eventually periodic? I have noticed that is a_{n}=f^{(n)}(x), the sequence a_{n} becomes eventually periodic for a multitude of f(x). If f(x) was a polynomial, and \... 7answers 7k views ### Function whose third derivative is itself. I'm looking for a function f, whose third derivative is f itself, while the first derivative isn't. Is there any such function? Which one(s)? If not, how can we prove that there is none? Notes: ... 6answers 31k views ### Proving that \lim\limits_{x\to\infty}f'(x) = 0 when \lim\limits_{x\to\infty}f(x) and \lim\limits_{x\to\infty}f'(x) exist I've been trying to solve the following problem: Suppose that f and f' are continuous functions on \mathbb{R}, and that \displaystyle\lim_{x\to\infty}f(x) and \displaystyle\lim_{x\to\... 2answers 5k views ### Why aren't integration and differentiation inverses of each other? Integration is supposed to be the inverse of differentiation, but the integral of the derivative is not equal to the derivative of the integral:$$\dfrac{\mathrm{d}}{\mathrm{d}x}\left(\int f(x)\... 5answers 4k views ### 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? 15answers 5k views ### 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 ... 8answers 3k views ### 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 ... 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$x=0$, ... 2answers 22k views ### 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 ... 1answer 4k views ### 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 ... 4answers 24k views ### 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? 6answers 7k views ### 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. ... 4answers 4k views ### 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. 5answers 5k views ### 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 ... 5answers 5k views ### 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 ... 6answers 1k views ### 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,...