# Tagged Questions

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

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### How to prove that $\log(x)<x$ when $x>1$?

It's very basic but I'm having trouble to find a way to prove this inequality $\log(x)<x$ when $x>1$ ($\log(x)$ is the natural logarithm) I can think about the two graphs but I can't find ...
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### Discontinuous derivative.

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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### 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 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 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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### Show that if $\lim_{x\to a}f'(x) = A$ then $f'(a)$ exists and equals A

I ran across this problem in my Analysis class and can't seem to come up with a good solution. Here's the question: Show that if $\lim_{x\to a}f'(x) = A$ then $f'(a)$ exists and equals $A$. $f$ is ...
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### Show that the function $g(x) = x^2 \sin(\frac{1}{x}) ,(g(0) = 0)$ is everywhere differentiable and that $g′(0) = 0$

Show that the function $g(x) = x^2 \sin\left(\frac{1}{x}\right) ,(g(0) = 0)$ is everywhere differentiable and that $g′(0) = 0$.
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I am trying to find a closed form solution for $\sum_{n\ge0} nx^n\text{, where }\lvert x \rvert<1$. This solution makes sense to me: $\sum_{n\ge0} x^n=(1-x)^{-1} \\ \frac{d}{d x} \sum_{n\ge0} x^n ... 3answers 237 views ### Failure of differential notation Through the informal use of differentials, the product rule can be "proved" by writing $$d(fg) = (f + df)(g + dg) - fg = df\,g + f\,dg + df\,dg.$$ Neglecting the product of two differentials, we ... 7answers 18k views ### Using the Limit definition to find the derivative of$e^x$I was wondering how we could use the limit definition $$\lim_{h \rightarrow 0} \frac{f(x+h)-f(x)}{h}$$ to find the derivative of$e^x$, I get to a point where I do not know how to simplify the ... 5answers 804 views ### Closed form for$n$th derivative of exponential of$f$What is the closed form for: $$\frac{\partial^n}{\partial x^n}\exp(f(x))=\exp(f(x))\cdot[????]$$ 2answers 453 views ### Proving that if$|f''(x)| \le A$then$|f'(x)| \le A/2$Suppose that$f(x)$is differentiable on$[0,1]$and$f(0) = f(1) = 0$. It is also known that$|f''(x)| \le A$for every$x \in (0,1)$. Prove that$|f'(x)| \le A/2$for every$x \in [0,1]$. I'll ... 6answers 304 views ### Are the any non-trivial functions where$f(x)=f'(x)$not of the form$Ae^x$This may seem like a silly question, but I just wanted to check. I know there are proofs that if$f(x)=f'(x)$then$f(x)=Ae^x$. But can we 'invent' another function that obeys$f(x)=f'(x)$which is ... 2answers 4k views ### What is the formula for nth derivative of arcsin x, arctan x, sec x and tan x? Are there formulae for the nth derivatives of the following functions? 1) arcsin x 2) arctan x 3) sec x, and 4) tan x Thanks. 1answer 1k views ### “Strong” derivative of a monotone function It is well known that if a function$f\colon \mathbb{R} \to \mathbb{R}$is monotone then$f'$exists almost everywhere. Is it true that if$f$is monotone then there exists (edit: I mean exists a.... 3answers 2k views ### Any ideas on how I can prove this expression? I don't have a lot of places to turn because i am still in high school. So please bear with me as i had to create some notation. In order to understand my notation you must observe this identity for ... 1answer 833 views ### Inverse of a bijection f is equal to its derivative Does there exist a differentiable bijection$f: \mathbb{R} \rightarrow \mathbb{R}$such that$f'(x) = f^{-1}(x)$? 1answer 767 views ### Partial derivative confusion. I don't understand partial derivatives. Here's an example that nails down my confusion: Suppose we have some variables$x$,$p$, and$q$with$p=x^2$and$q=e^x$. Then $$\frac{\partial q}{\partial p} ... 3answers 143 views ### Proving a function is constant, under certain conditions? The problem: Assume f: \mathbb{R} \rightarrow \mathbb{R} satisfies |f(t) - f(x)| \leq |t - x|^2 for all t, x. Prove f is constant. I believe I have some intuition about why this is the case; ... 6answers 3k views ### Uses of \lim \limits_{h\to 0} \frac{f(x+h)-f(x-h)}{2h} I have been wondering whether the following limit is being used somehow, as a variation of the derivative:$$\lim_{h\to 0} \frac{f(x+h)-f(x-h)}{2h} .$$Edit: I know that this limit is defined in ... 3answers 394 views ### Deriving the Normalization formula for Associated Legendre functions: Stage 1 of 4 The question that follows is needed as part of a derivation of the Associated Legendre Functions Normalization Formula:$$\color{blue}{\displaystyle\int_{x=-1}^{1}[{P_{L}}^m(x)]^2\,\mathrm{d}x=\left(\... 4answers 281 views ### Verify$y=x^aZ_p\left(bx^c\right)$is a solution to$y''+\left(\frac{1-2a}{x}\right)y'+\left[(bcx^{c-1})^2+\frac{a^2-p^2c^2}{x^2}\right]y=0$In order for the question that I have to make any sense I must first include some background information as given in my textbook: The standard form of Bessel's differential equation is $$x^2y^{\... 2answers 353 views ### \alpha-derivative (concept) I found the following definition: Given an real number \alpha, we say that a function f: \mathbb{R} \rightarrow \mathbb{R} is \alpha-differentiable at 0 if exists the limit:$$\lim_{t \to 0^+} ... 2answers 964 views ### Continuous function with linear directional derivatives=>Total differentiability? As in the title: If$f\colon\mathbb{R}^n\to\mathbb{R}$is continuous in$x$and has directional derivatives$\partial_vf(x)=L(v)\,\forall v\in\mathbb{R}^n$, where$L$is linear, does this imply that$...
Suppose we have a continuous function $f : \mathbb{R} \to \mathbb{R}$. Suppose also that for a certain point $c$, $\lim_{x \to c} f'(x)$ exists. Must $f'(c)$ exist as well, and be equal to this limit? ...