# 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 do I show $f(x+2)-f(x)>2 \forall x$?

For the function $f(x)=x\cos{\frac{1}{x}}$, $x\geq1$, How do I show that $f(x+2)-f(x)>2 \forall x$?
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### What to choose $g(x)$ as so that $f′′(x)−2f′(x)+f(x)≥e^x$?

Let $f:[0,1]→R$ (the set of all real numbers) be a function. Suppose the function $f$ is twice differentiable, $f(0)=f(1)=0$ and satisfies $f′′(x)−2f′(x)+f(x)≥e^x$, $x∈[0,1]$.Prove that ...
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### derivative of many roots

if $$y=\frac{(1+2x)^{1/2}.(1+4x)^{1/4}.(1+6x)^{1/6} ... (1+100x)^{1/100}}{(1+3x)^{1/3}.(1+5x)^{1/5}.(1+7x)^{1/7} ... (1+101x)^{1/101}}$$ then find y' at x=0 Already tried to find a general term ...
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### Deriving the wave equation in 3 dimensions and the history of it

I'm trying to find how the wave equation was derived in 3 dimensions. Surprisingly, there isn't much information available on this apart from wikipedia of all places ...
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### If the second derivative of a function is zero, why is the second derivative test inconclusive?

2nd derivative test gives three possibilities: 1) greater than zero (strict local min) 2) less than zero (strict local max) 3) equal to zero - no information It is this third case that I do not ...
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### If $\sin y=a\sin(x+y)$ prove $\frac{\rm d y}{\rm d x}=\frac{\sin a}{1- 2x\cos a +x^2}$

If $\sin y=a\sin(x+y)$ prove $\frac{\rm d y}{\rm d x}=\frac{\sin a}{1- 2x\cos a +x^2}$ I am not finding any proper way even to express $y$ only in terms of $x$ too which could reduce bit ...
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### Holder condition and differentability

Prove that if a function is $C^2$ on a closed interval, then it satisfies holder condition of order 2. Thanks
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### What should I know about half vectorization and Kronecker product to do this matrix differentation?

I have a scalar function as follows: \begin{equation*} \ell(\beta, \Sigma, \mu, \Lambda) = \sum_{i=1}^{m} \left[\boldsymbol{y}_{i}^{T} \left(X_{i}\beta + Z_{1} \mu_{i} \right) - ...
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### How to verify this identity?

From Weinstock, "Calculus of Variations", p.24: We have the readily verifiable identity \begin{align}\frac{d}{dx}\left(y'\frac{\partial f}{\partial y'}-f\right) = ...
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### To check continuity and differentiability

Consider the function I am having problem with checking continuity because of y. Regarding differentiability i can apply Leibniz rule to get explicit formula.But then modulus part troubles me. Can ...
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### Where is the function series $f(x)=\sum\limits_{n=0}^\infty\frac{e^{-nx}}{n^2+1}$ differentiable?

I was asked to analyze the convergence, continuity and differentiability intervals of the next function series: $f(x)=\sum\limits_{n=0}^\infty\frac{e^{-nx}}{n^2+1}$ I already know that this ...
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### Differentiate with respect to $x^{a}(t)$

having a little difficulty with this conceptually. Can someone quickly walk through this differentiation please? $K=\frac{1}{2}g_{ab}\dot{x}^{a}\dot{x}^{b}$ Find $\frac{dK}{dx^{a}}$ In this case, ...
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### What do instantaneous rates of change really represent?

The derivative of $f(x)$ is the value of the limit of the average rate of change of $y$ with respect to $x$ as the change in $x$ approaches $0$. This is the value, in other words, that the average ...
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### Problem with a notation of symbolic derivatives.

Let's say we have got a function $F(G(B)\cdot C)$, i.e function $F$, which is a function of a function $G$ and variable $C$; also function $G$ is a function of variable $B$. Now I want to obtain the ...
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### Maths for economics: finding the level of production that minimises marginal cost [closed]

Let the total cost function of a firm be given by: $$TC(Q)= 16Q^3 - 72Q^2 + 446Q + 90$$ Find the level of production that minimises the marginal cost of production. (This is basically taking the ...
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### Is it true that $\lim_{h \to 0} \frac{f(x_0 + h, y_0 + h) - f(x_0 + h, y_0)}{h} = \frac{\partial f}{\partial y}(x_0, y_0)?$

let $f: \mathbb{R}^2 \to \mathbb{C}$ be a continuous function. Let $(x_0,y_0) \in \mathbb{R}^2$ be a point in $\mathbb{R}^2$ such that both partial derivatives $\frac{\partial f}{\partial x}$ and ...
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### Proof for $\frac{2}{\pi}x \lt \sin{x}$ for $x \in (0,\frac{\pi}2)$

The following is part of exercise 6.26.21 from Tom Apostol's Calculus Volume 1. I wonder if my proof is correct and if there is a simpler alternative proof. Prove the following by examining the ...
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### Is $f$ differentiable at $(0,0)$?

Define $f:\mathbb{R}^2\rightarrow \mathbb{R}$ by $f(x,y)=\displaystyle \frac{y^3-\sin^3x}{x^2+y^2}$ if $(x,y)\neq (0,0)$ and $f(0,0)=0$. My question is, is $f$ differentiable at $(0,0)$? First ...
### Does Darboux theorem imply that $f'$ cannot have jump discontinuity?
Does the Darboux theorem for derivatives imply that a derivative on a interval $I$ cannot have jump discontinuity? Darboux theorem states that the derivative function follow the intermediate value ...