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

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0
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5answers
105 views

Prove that $e^{2x+1}+e^{x}+1/4$ is always greater than 0

Prove that $e^{2x+1}+e^{x}+1/4$ is always greater than 0 I was thinking of supposing that the function is equal to or less than 0, and showing that the supposed equation would not work out. How would ...
1
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2answers
60 views

Why does $\mathrm{d}l=v\,\mathrm{d}t$ imply that $\frac{\mathrm{d}}{\mathrm{d}t}=v\frac{\mathrm{d}}{\mathrm{d}l}$?

If we have a lengthening pendulum and the length $l$ of the pendulum at time $t$ is $$l=l_0+vt$$ where $l_0$ is the initial length of the pendulum and $v$ is the velocity for which the pendulum's ...
0
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2answers
56 views

Derivative of $\frac{(x\cos\,\theta-y\sin\,\theta)^2}{a^2}+\frac{(x\sin\,\theta+y\cos\,\theta)^2}{b^2}=1$

The derivative of $$\frac{(x\cos\,\theta-y\sin\,\theta)^2}{a^2}+\frac{(x\sin\,\theta+y\cos\,\theta)^2}{b^2}=1$$ is $$\frac{\mathrm dy}{\mathrm dx}=-\frac{a^2 x\,\sin^2\theta+y(a-b)(a+b)\sin\,\theta ...
4
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2answers
57 views

A sufficient condition on $C^1$ positive functions for $f(x+y)<f(x)+f(y)$

I am trying to show that if $f:(0,+\infty)\rightarrow\mathbb R$ is a $C^1$ function such that $$f'(x)<\frac{f(x)}{x}\quad \forall x\in (0,+\infty) \tag{$\star$}$$ then $$f(x+y)<f(x)+f(y)$$ ...
1
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1answer
33 views

General form of function given that it is differentiable

Suppose a function satisfies the relation $f(x+y)=f(x)+f(y)$. How can I show that the more general form of $f(x)$ is $f(x)=kx$ where $k$ is a constant given that $f$ is differentiable?
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3answers
74 views

Find the 8th derrivative of the function $h(x) = xe^x $using sequences

How do you find the 8th derivative of $h(x) = x e^x $ without doing it "manually". I know that $\displaystyle e^x = \sum_{i=0}^n \frac{x^n}{n!} $ so that $\displaystyle h(x) = x \sum_{i=0}^n \frac{x^...
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2answers
34 views

Determine whether $f''(x)>0$ or not in a given interval

Let $f:[0,4]\rightarrow\mathbb{R} $ be a twice differentiable function.Further,let $f(0)=1,f(2)=2,f(4)=3$. Then which of the following can be regarded as true? $A:$there does not exist any $x_1\in(0,...
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0answers
40 views

Getting along with differential of a function, Leibniz notation

$\frac{dx}{dy}$ is one of worst things man can occur in mathematics. This conclusion can be supported by Wikipedia article on it. For example: Augustin-Louis Cauchy (1823) defined the differential ...
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0answers
11 views

cusps of bivariate polynomial equation

I plotted an implicit curve: $$f(x,y)=\sum_i^N a_i x^i y^{N-i}=0 \;\;\;a_i\in R$$ and found that the curve has sharp corners (cusps). (area where f>0 are painted red, otherwise green) If this is ...
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2answers
53 views

Find the value of the constant k that makes the function continuous

h(x) = \begin{cases} x^{2} & \text{if $x\le5$} \\ x+k & \text{if $x>\ 5$} \\ \end{cases} Answer choices are A. k=20 B.k=-5 C. k=5 D. k=30
2
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2answers
21 views

Derivative that includes several functions of time

I'd like to compute the following derivative (i.e., solve for $v$): \begin{align} \frac{dv}{dt} = \frac{-v(t) + I_{rec}(t) + I_{ext}(t)}{\tau_m}. \end{align} I know that if I had $\frac{dv}{dt}=\...
2
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0answers
17 views

Extension to Leibniz integral rule

I want to differentiate an integral, where things are a bit more complicated than the standard Leibniz integral rule. The integral is shown below: $\frac{d}{dz} \int_a^b f(y(x),z(x)) dx$ Here the ...
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1answer
13 views

Orthogonality of $\nabla f(x_0)$ and $\gamma'(0)$ where $\gamma(0)=x_0$

let $$f:\mathbb R^n\rightarrow \mathbb R$$ be differentiable, $c\in \mathbb R,\epsilon >0,x_0\in \mathbb R^n$ and $\gamma:(-\epsilon,\epsilon)\rightarrow \mathbb R^n$ a continuous differentiable ...
0
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1answer
40 views

Show that $f(x)=f(y)$ under certain conditions

Let $f,F:\mathbb R^3\rightarrow \mathbb R$ be two functions such that $f\in C^1$ and the equality $$\triangledown f(x)=F(x)x^t\hspace{0.7cm}\forall x\in \mathbb R^3$$ holds. I have to show that in ...
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3answers
27 views

Value of differentiation at a given point.

If $x^y\cdot y^x=16$ then $\dfrac{dy}{dx}$ at $(2,2)$ is ?. After calling equation as $f(x)$ and differentiating I get $yx^{y-1}\cdot y^x+x^y\cdot y^x\cdot\ln(y)$ after plugging in value I get $16(1+\...
3
votes
2answers
118 views

A discontinuous function with smooth sections

I am searching for $f : U\rightarrow \mathbb R $ defined in an open square $U$ in $\mathbb R^2$ so that $(0,0) \in U$, $f$ is not continuous at $(0,0)$, for each $x$ the function $y\mapsto f(x,y)$ is ...
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1answer
21 views

Finding coordinates at which tangent lines intersect

Tangent lines are drawn to the function $f(x)=x^2-4/x$ at the points $(-1,5)$ and $(1,-3)$. Find the coordinates of the point at which the tangent lines intersect. I'm not sure how to approach this ...
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2answers
58 views

Finding derivative of $\frac{1}{\sqrt{x+2}}$ using only the definition of the derivative $f'(x) = \lim_{h \to 0} \frac{f(x + h) - f(x)}{h}$

I need to find $\frac{d}{dx}(\frac{1}{\sqrt{x+2}})$ only using the basic definition of the derivative $f'(x)=\lim_{h \to 0}\frac{f(x+h)-f(x)}{h}$, but I'm not having any luck with the algebra. Any ...
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3answers
59 views

proof of the second symmetric derivative

prove that is $f''(a)$ exists, then $$f''(a)=\lim_{h\to0}\frac{f(a+h)+f(a-h)-2f(a)}{h^{2}}$$ I really have no idea on this one. Am I supposed to apply the mean value theorem?
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0answers
9 views

Infinite differentiability of a (cumulant generating) function

Suppose that I have a moment generation function $M(t)$ that is positive and finite for all $t\in\mathbb{R}$. I know that this implies that $M(\cdot)$ is infinitely differentiable on $\mathbb{R}$. ...
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3answers
29 views

Finding the stationary points of the curve.

Find the coordinates of the stationary points on the curve $y=3 xe^{2-x^2}$. I went ahead and used the product rule to get my derivative, which would later give me the $x$-coordinates. My derivative ...
1
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3answers
38 views

What is $\frac{\mathrm{d}f(x,x)}{\mathrm{d}x}$

Related: Is $\frac{\partial}{\partial x} f(x,y(x))$ ambiguous? If $f(x,x) = x^2$, is it correct to say that $\dfrac{\mathrm{d}f(x,x)}{\mathrm{d}x} = 4x$? I know that $\dfrac{\mathrm{d}f(x,y)}{\...
0
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1answer
14 views

Use derivatives of two functions to make statements about their respective rates of growth?

In general, whether $\lim_{x \to \infty} \frac{f(x)}{g(x)} = 0$, or $\lim_{x \to \infty} \frac{f(x)}{g(x)} = \infty$, or $\lim_{x \to \infty} \frac{f(x)}{g(x)} = C$, would the same be true using any ...
1
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1answer
46 views

gradient of trace$(ABA^TC)$ w.r.t a Matrix A.

With n-order Matrix A,B,C.I was trying to find $ \nabla_A trace(ABA^TC)$ This answer:Proof for the funky trace derivative : $d (\operatorname{trace} (ABA'C))$? suggested: $$ \nabla_A \...
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2answers
36 views

Can chain rule be used in first step

I was wondering if it were possible to use the chain rule in the first step to differentiate the f.f.g function: $$f(x) = (1 + \sin x)^{\cot x}$$ I know the obvious first step is to use the power ...
1
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1answer
59 views

Can the nabla symbol be used for a covariant derivative?

Can $$ \nabla_{\nu} A^{ \mu\nu} $$ represent a covariant derivative with respect to $\nu$? If not what can it be? I'm reading a textbook on General Relativity, and such operations appear without any ...
1
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1answer
32 views

Derivative of convolution type integral equation with respect to time $ x(t) = \int\limits_{0}^t \gamma^{t-s} u(s)ds$

Consider the equation: $$x(t) = \int\limits_{0}^t \gamma^{t-s} u(s)ds$$ where $x, u$ are functions and $\gamma \in \mathbb{R}_{>0}$ I am trying to obtain the derivative of this equation with ...
0
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0answers
18 views

Total derivative using $(x,y)\mapsto f(x,y)$ notation

Related to What is the difference between $\frac{\mathrm{d}}{\mathrm{d}x}$ and $\frac{\partial}{\partial x}$? Is it possible to make sense of $(x,y)\mapsto f(x,y)$ when taking the total derivative ...
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0answers
40 views

composition with biholomorphism

Let $f:G_1 \rightarrow G_2$ be a local biholomorphism and $g:G_2 \rightarrow G_1 $ continious such that $(f\circ g)(z)=z$ Proof that $g$ is a biholomorphism I know that $g$ is biholomorphic if $g' \...
1
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1answer
47 views

Derivatives of the Dirac delta function

From what I understand the Dirac's Delta derivatives have the meaning $$\int_{-\infty}^{\infty}\delta^{(k)}(x)\phi(x)dx=(-1)^k\int_{-\infty}^{\infty}\delta(x)\phi^{(k)}(x)dx$$ Assuming, of course that ...
0
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5answers
63 views

First Year Calculus Problems

I'm doing a few past exam papers for my calculus test. I came across a few problems which I thought would be worth asking about. 1.) Consider the function $f(x) = \begin{cases} x+1, & \text{if ...
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1answer
21 views

Error term between $f(x)$, its average value and value at midpoint

Let $f$ be a smooth function on interval $[a,b]$. Define the average $\bar{f}=\dfrac{1}{b-a}\int_a^bf(y)\,dy$ and $\bar{x}=\dfrac{a+b}{2}$, then for any $x\in [a,b]$, we can write $$f(x)-\bar{f}=c(x-\...
12
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5answers
287 views

What is the difference between $\frac{\mathrm{d}}{\mathrm{d}x}$ and $\frac{\partial}{\partial x}$?

Is there not any difference between $\frac{\mathrm{d}}{\mathrm{d}x}$ and $\frac{\partial}{\partial x}$ as long as your function has one variable? $f(x) = x^3\implies \left\{\begin{align}&\dfrac{\...
0
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0answers
56 views

How to show that $x/(1+|x|)$ is not differentiable at $x = 0$?

$$f(x) = \frac{x}{1+|x|}$$ $$\Rightarrow f'(x) = \frac{1}{(1+|x|)^2}$$ I understand from inspection of the graph that $f(x)$ is not differentiable at $x = 0$. (EDIT: Actually graph looks OK, is my ...
0
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1answer
36 views

Discontinuity and Dirac's Delta Function

Can someone help me understand how he came up with Dirac's function to differentiate that discontinuous periodic function? I am familiar with Dirac's function, but I don't understand where it came ...
0
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1answer
29 views

Fixed point analysis in the Wilson-Cowan model

i guess this is a rather simple question, but given my non-mathematical background, i'm a bit stuck. i'm trying to find the jacobian matrix for the follwing dynamical system (wilson-cowan model). the ...
0
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1answer
25 views

Interpretation of the Derivative of a Quaternion

Considering this definition of the derivative of a quaternion: $$dq/dt = 1/2 w q$$ If we're considering $q$ to be a unit quaternion representing an orientation in 3D with $(cos(theta/2), sin(theta/2)*...
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3answers
59 views

Given a rocket with constant acceleration after t = 4, when will it hit the ground?

A rocket is launched straight upward. During the first four seconds of powered flight, its height is given by: $h(t) = 16.1t^2 − 1.75t^3$ The function is valid when $0 ≤ t ≤ 4$ $t$ in seconds and $...
0
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1answer
27 views

Finding acceleration of an object given height

The height of a moving object is given as a function of time. $$h(t) = 3.0 + 2.7 \sin(1.3t + 0.9)$$ $t$ is measured in seconds and $h$ is measured in feet. Given this, I've found the velocity and ...
9
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3answers
302 views

Why not Define Infinite Derivatives?

Is there any particular reason "infinite" derivatives are not well-defined? For example, $x \mapsto x^{\frac 13}$ at $x=0$. More precisely, what is wrong with the following definition of ...
0
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1answer
23 views

Derivatives relation

Please, help me solve the following problem: Suppose $$\frac{df(x,y)}{dx}>0, \qquad \frac{df(x,y)}{dy}<0, \qquad \frac{d^2f(x,y)}{dxdy}<0$$ Is it true that if $y_{2}>y_{1}>0$ then $...
0
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1answer
25 views

How to solve this optimization question with the Extreme Value Theorem?

Consider the region in the x-y plane that is bounded by the x-axis and the function $f(x)=b-ax^2$. Construct a rectangle whose base lies on the x-axis and is centered at the origin, and whose sides ...
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1answer
20 views

Rectangular prism optimization using extreme values

A box with a rectangular base, whose length is twice its width, is to have a closed top. The area of the material in the box is to be $192in^2$. What should the dimensions of the box be in order to ...
0
votes
1answer
105 views

Calculating this derivative

I'm currently having trouble to verify this fact on page 90 in Robert L. Devaneys "An Introduction to Chaotical Dynamical Systems": Let $F(x,\lambda) = f_\lambda (x)$. Assume $f_\lambda (x) = 0$ for ...
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1answer
48 views

Partial Derivative of Gaussian function: Matrix differentiation

I am interested in partial derivative of the following term w.r.t $x_1$ $$\mathbf g = \begin{bmatrix}x_1-k_1 ,& x_2-k_2, & x_3-k_3 \end{bmatrix} \begin{bmatrix}s_{11}& s_{12} & s_{13}...
0
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0answers
16 views

Why is $\left\lvert D^{\alpha}\left(\frac{D\phi}{|D\phi|^2}(\xi)\right)\right\rvert\le C_{\alpha}|\xi|^{-1-|\alpha|}$

Why is $$\left\lvert D^{\alpha}\left(\frac{D\phi}{|D\phi|^2}(\xi)\right)\right\rvert\le C_{\alpha}|\xi|^{-1-|\alpha|}$$ for $\phi\in C^{\infty}(\mathbf R^n)$ sucht that $D\phi(0)=0$, $D\phi\neq0$ if $\...
1
vote
1answer
42 views

Differentiability of a function $f:\mathbb R^2\rightarrow \mathbb R$

I want to prove that the funktion $f:\mathbb R^2\rightarrow \mathbb R,\hspace{0.5cm} f(x,y)=\begin{cases} (x^2+y^2)\sin\big(\frac{1}{\sqrt{x^2+y^2}}\big),& \text{if } (x,y)\neq (0,0)\\ 0,...
10
votes
2answers
166 views

Prove that $\sin x \cdot \sin (2x) \cdot \sin(3x) < \tfrac{9}{16}$ for all $x$

Prove that $$ \sin (x) \cdot \sin (2x) \cdot \sin(3x) < \dfrac{9}{16} \quad \forall \ x \in \mathbb{R}$$ I thought about using derivatives, but it would be too lengthy. Any help will be ...
0
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0answers
32 views

How to derivative the function which have constraint like $x^2+y^2+z^2 = 1$

For example , we have function like $f(x,y,z) = 1 - 2\times x^2 + y + z$ with constraint $x^2+y^2+z^2 = 1$ when I need to compute the derivative of $\frac{∂f}{∂x}$ , should it be $\frac{∂f}{∂...
6
votes
4answers
271 views

Verify $y=x^aZ_p\left(bx^c\right)$ is a solution to $y''+\left(\frac{1-2a}{x}\right)y'+\left[(bcx^{c-1})^2+\frac{a^2-p^2c^2}{x^2}\right]y=0$ Method #1

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^{\...