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

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2
votes
3answers
80 views

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 $f(x+2)-f(x)>2 \forall x$?
1
vote
1answer
40 views

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 ...
2
votes
4answers
222 views

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 ...
1
vote
0answers
42 views

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 ...
1
vote
1answer
51 views

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 ...
2
votes
3answers
69 views

I would like to get the derivative of this function: $ f(x)=(x-a)^2(x-b)^2 $ and $f(x) = \frac{1}{e}$

I want to get the derivative of this function: $$ f(x)=(x-a)^2(x-b)^2 $$ for $x ∈< a, b >$, $$f(x) = \frac{1}{e}$$ for all other $x$. Now I know the result is: $$ f'(x) = 2(x − a)(x − b)(2x − ...
9
votes
1answer
70 views

An Odd Mean Value Theorem Problem

If $f: [x_1,x_2] \to \mathbb{R}$ is differentiable, show for some $c \in (x_1,x_2)$ that $$ \frac{1}{x_1-x_2} \left| \begin{matrix} x_1 & x_2 \\ f(x_1) & f(x_2) \end{matrix} ...
2
votes
1answer
48 views

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 ...
0
votes
1answer
34 views

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
0
votes
1answer
86 views

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) - ...
2
votes
2answers
48 views

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) = ...
1
vote
4answers
56 views

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 ...
2
votes
1answer
65 views

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 ...
0
votes
1answer
22 views

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, ...
3
votes
5answers
262 views

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 ...
2
votes
0answers
29 views

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 ...
4
votes
2answers
47 views

Sum of partial derivatives

Suppose that $$\mu_i(x)=x_i \int_0^1 t^{n-1} \rho(tx) dt$$ where $\rho$ is a function on $\mathbb R^n$ and $tx=(tx_1,\dots,tx_n)\in \mathbb R^n$. Show that $$\sum_{i=1}^n ...
0
votes
3answers
42 views

How many points of maximum/minimum in the range $[-2,2]$?

An exercise asks me to find the points of maximum and minimum of the following function. $$f(x)=(x^2-2x)e^x$$ in the following range $$[-2,2]$$ After finding the first derivative $$f^1(x)=e^x(x^2-2)$$ ...
2
votes
2answers
40 views

Prove the following inequality (probably) using derivatives

In chapter where we use derivatives for determining local minima/maxima there is this inequality where I really do not know where to start. Prove: $$ \frac{1}{2^{p-1}} \leq x^p +(1-x)^p \leq 1 $$ ...
8
votes
2answers
166 views

Why can you mix Partial Derivatives with Ordinary Derivatives in the Chain Rule?

This question is a simplified version of this previous question asked by myself. The following is a short extract from a book I am reading: If $u=(x^2+2y)^2 + ...
1
vote
2answers
62 views

$f$ ia continuously differentiable function with $f'(c)=0$…

Let $f$ be a continuously differentiable function on $[a,b].$ There is a number $c$ in $(a,b]$ such that $f'(c)=0.$ Then prove that there is a fixed number $\xi\in(a,b)$ such that ...
0
votes
1answer
83 views

How to properly take derivative of discrete data numerically in matlab?

Taking derivative of discrete data requires some fitting and then using the b-form of the polynomial. How to make a 'good' fit and properly take derivative? I am confused what is the right behavior ...
0
votes
1answer
37 views

Find the point on the graph of $g(x) = 2x^2 + 3x + 1$ at which the normal line is parallel to the line with equation $x - y = 2$

I suspect I should use implicit differentiation at some point to solve this problem but I'm somewhat clueless on how to approach this problem. Would appreciate tips and directions
1
vote
3answers
48 views

How do I calculate the gradient of a discrete function?

In the continuous case, I have $$\lim_{x\to x_0} \frac{f(x) - f(x_0)}{x - x_0} = \lim_{h\to 0} \frac{f(x_0 +h) - f(x_0)}{h}$$ But what is the gradient of the function $f: \mathbb{N} \rightarrow ...
3
votes
2answers
59 views

Is finding the second derivative of $\sqrt[3]{\vert x\vert}$ the best method to determine if it is convex?

I have an exercise where I have to tell on which intervals a function is concave or convex. I usually do it using second derivative, but I would like to know if there is a simpler way of doing so, ...
2
votes
0answers
24 views

Is there a similarity solution for this PDE? (with discussion, kindly check)

I have a PDE for $h(x,t)$ of this form $$h_t+Ah^{-1}+(h^3h_x)_x+Bh_{xx}+(h^3h_{xxx})_x=0,$$ where the subscripts denote the partial derivatives, and $A$ and $B$ are all constants. I'm wondering ...
4
votes
1answer
115 views

With the exception of the Total Derivative; Is the Chain Rule valid when it has partial derivatives mixed with ordinary derivatives?

I know that the chain rule for a function of one variable $y=f(x)$ is written as $$\frac{{\rm d}}{{\rm d}x}=\frac{{\rm d}}{{\rm d}y}\times \frac{{\rm d}y}{{\rm d}x}\tag{1}$$ I also know that if ...
1
vote
1answer
48 views

Mapping tangent vectors between differential manifolds

Nigel Hitchen, in his notes on differential manifolds gives a definition of the derivative of a smooth map between two manifolds that appears to assume the the following assertion is self-evident: ...
0
votes
1answer
35 views

If $f(x)=x^3-3x^2+x$ and $g(x)$ is the inverse of $f(x)$, find $g'(x)$

I found this question in my text book: If $f(x)=x^3-3x^2+x$ and $g(x)$ is the inverse of $f(x)$, find $g'(x)$ I know that: $$[f^{-1}(x)]'=\frac{1}{f'(f^{-1}(x))}$$ But, this function is not one ...
0
votes
0answers
23 views

Function of class $C^k$ in the Frechet sense

There is a theorem that states that $f: \mathbb{R}^n \to \mathbb{R}^m$ is of class $C^1$ iff every partial derivative $\frac{ \partial f_i}{ \partial x_j}$ is continuous. Can this be generalized for ...
2
votes
3answers
49 views

Derivative of dot product vs derivative of scalars

Suppose $\vec{v}(t)$ is the velocity (vector) function. Then: $$\frac{\mathrm{d} (\vec{v}\cdot\vec{v})}{\mathrm{d} t}=2 \vec{v} \cdot \frac{\mathrm{d} \vec{v} }{\mathrm{d} t}=2 \vec{v} \cdot ...
2
votes
2answers
43 views

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 ...
2
votes
1answer
41 views

Prove multi-dimensional Mean Value Theorem

I've been asked to prove multi-dimensional Mean Value Theorem. I'd be grateful if someone could give me feedback if it is okay. Proof of Mean Value Theorem: Let $f: [a,b]\rightarrow ...
32
votes
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 ...
0
votes
1answer
61 views

Implications of the existence of $\lim_{x \to a+} \frac{f'(x)}{g'(x)}$

I'm reviewing my calculus notes of a variant of L'Hospital's rule and there's one step in the proof that I found puzzling. The theorem states: Let $a \in \mathbb{R}, \lim_{x \to a+} f(x) = \lim_{x ...
2
votes
1answer
40 views

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 ...
0
votes
2answers
77 views

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 ...
1
vote
2answers
74 views

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 ...
0
votes
1answer
9 views

How to compute this partial derivative with scalar product?

Let $\langle u,v\rangle =\int_0^1 u(x)v(x)\, dx$. I would like to compute $$ L(\langle y(x,t),\Phi(x)\rangle\Phi(x)), \text{ where }~~L=\frac{\partial^2}{\partial x^2}. $$ My computation gave $$ ...
5
votes
2answers
61 views

What does is the meaning of $\frac{d}{dx}+x$ in $(\frac{d}{dx}+x)y=0$?

I believe understand the meaning of the infinitesimals $dx$ and $dy$. I understand that $dx/dy$ is the ratio of an infinitely small change in $x$ to an infrequently small change in $y$. However, I can ...
1
vote
0answers
28 views

Chain rule in $\mathbb{R}^d$, with $d\ge 2$.

Given $\Omega\subset\mathbb{R}^d$ be an open bounded set with Lipschitz boundary, let $v\in (H^1(\Omega))^d$, $\psi\in H^1(\Omega)$ $T_K(x):=B_K x+b_K$, where $B_K$ is a non-singular invertible ...
1
vote
1answer
59 views

Banach fixed point theorem - Find $x$ such that $f(x) = 0$.

Let $f: \mathbb R \to \mathbb R$ be a continuous function with a continuous derivative. In short $f \in C^1$. We know that $0<c\leq f'(x) \leq d < \infty$. We want to prove that $\exists! x_0 ...
1
vote
3answers
62 views

Find $G'(2) $, where $ G(x)= \int_{\pi/x}^{\pi x}\cos^{15}t\,dt$ [closed]

Find $G'(2) $ where $$ G(x)= \int_{\pi/x}^{\pi x}\cos^{15}t\,dt$$ Honestly, I don't know where to start. I need a full and detailed answer.
1
vote
0answers
27 views

Take the derivative of the parameter in an integral.

Consider two smooth functions $f(x)$, $g(x)$ defined on $\mathbb{R}^n$, and $g$ has compact support. Easily, we have $$ \int_{\mathbb{R}^n}f(ax)g(x)dx=\int_{\mathbb{R}^n}a^{-d}f(x)g(a^{-1}x)dx, $$ ...
9
votes
2answers
264 views

What is the derivative of $x!^{x!^{x!^{x!^{x!^{x!^{x!^{.{^{.^{.}}}}}}}}}}$

What is the derivative of $$x!^{x!^{x!^{x!^{x!^{x!^{x!^{.{^{.^{.}}}}}}}}}}$$ My effort: Let $$g(x)=x!^{x!^{x!^{x!^{x!^{x!^{x!^{.{^{.^{.}}}}}}}}}}=>g(x)=x!^{g(x)}$$ Taking natrual log on both ...
10
votes
2answers
286 views

Does a smooth “transition function” with bounded derivatives exist?

Does there exist a function $f: \mathbb{R} \to \mathbb{R}$ having the following properties? $f(x) = 0$ for all $x \le 0$. $f(x) = 1$ for all $x \ge 1$. For $0 < x < 1$, $f$ is strictly ...
0
votes
1answer
61 views

Show that the roots of P' lie in the same half plane as the roots of P

The problem statement is: Part(a) Assume P(z) is a non-constant polynomial with all of its roots in some half plane H. Show that the derivative P'(z) must also have all of its roots in H. Part (b) ...
2
votes
3answers
36 views

Find the derivative $dy/dx$ from the parametric equations for $x$ and $y$

Let \begin{cases} y=2t^2-t+1 \\ x=\sin(t) \end{cases} find $\frac{dy}{dx}$ Is this all that I need to do? $$\frac{4t-1}{\cos(t)}$$
0
votes
1answer
78 views

Growth rates slower than logarithmic? [closed]

So far, I've been able to determine growth rates using the following limit:$$\lim_{x\to\infty}\frac{f(x)}{g(x)}$$Which, if need be, can be solved with calculus. From this, I deduced that it is very ...
2
votes
2answers
68 views

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 ...