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

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1answer
42 views

Find series of real functions for which the sum has continuity properties

The sequence of continuous real functions $f_i$ is defined on the unit interval $[0, 1]$. Each $f_i$ is composed of finitely many linear segments, each segment has slope +1 or −1, moreover $f =\sum ...
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0answers
56 views

Partial Derivatives Calculus

If $$x^x\cdot y^y\cdot z^z=c$$ then prove that $$\frac{\partial^2z}{\partial x \, \partial y}=(-x\log_ex)^{-1}$$ How to do this? I was first taking logarithm and then differentiating but did not work. ...
2
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1answer
44 views

When there exists function $f$ such that for given $g$ we have $f'=g$?

I am looking for a theorem that states when function $g: \mathbb R \mapsto \mathbb R$ is a derrivative, i.e. there exists $f$ such that $f'=g$. What about if we just need this condition almost ...
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4answers
131 views

Why this function is continuous and not differentiable at point $x=1$

I have a function $$f(x) = \begin{cases}x^2+2,& x\leq 1\\x+2 ,& x > 1\end{cases}$$ I have to show that this function is continuous and not differentiable at point $x=1$, but when I look for ...
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2answers
51 views

$n$-th derivative of $f(\ln x)$

Find general formula for $n$-th derivative of $y = f(\ln x)$. To start with I found couple of derrivatives: \begin{align} y' &={1 \over x}f'(\ln x) \\ y'' &={1 \over x^2}(f''(\ln x)-f'(\ln ...
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1answer
31 views

Proving differentiability on every interval equals proving differentiability on all of $\mathbb{R}$?

The question actually came to me from series of functions. Suppose $f_n(x)$ is a sequence of continuous differentiable functions, that $$\sum_{n=1}^\infty f_n'(x)$$ converges uniformly on any closed ...
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1answer
35 views

Application of Euler Theorem On homogeneous function in two variables.

Euler theorem says, If $$u=f(x,y)\text{ ,homogeneous}$$ Then,$$x\frac{\partial u}{\partial x}+y\frac{\partial u}{\partial y}=nu$$ Where $$n\to \text{degree of function}$$ Question If ...
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3answers
40 views

Definition of derivative $f(x) = 3x - \frac{1}{x^2}$

$f(x) = 3x - \frac{1}{x^2}$ I am finding this problem to be very tricky:
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2answers
63 views

Find the derivative for $f(x) = \sqrt{x+2}$ by definition

Use the definition of derivatives to show that the derivative of $f(x) = \sqrt{x+2}$ is $$ f'(x) = \frac12(x+2)^{-\frac12} $$ My proof so far: Given $\epsilon > 0$ , $\exists ...
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1answer
40 views

Total derivative and chain rule

I am aware this could be a dumb question but I got myself stuck in some complicated calculus that I have greatly simplified to focus on my lack of understanding. My goal is to compute the derivative ...
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0answers
31 views

Find for which values of C a function is differentiable

Find for which values of $C \in \mathbb{R}$ the function $f$: $\mathbb{R^2} \to \mathbb{R}$ is differentiable, with $f$ defined by: $$f(x,y) = \begin{cases} \frac{|x|^C y}{\sqrt{x^2 +y^2}} ...
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1answer
27 views

Derive a power series expansion about $x=0$

$(1+x)^2y''(x) - xy'(x) + y(x) = e^{-x} $ If possible, derive a power series expansion about $x=0$ up to the first 3 terms. State which parts are the particular solution and which parts are the ...
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1answer
27 views

Computing the derivative: $\frac{\partial}{\partial x} \left\{ \int_0^t \int_{x - t + \eta}^{x + t - \eta} F(\xi,\eta) \,d\xi\, d\eta \right\}$

Let's say that $F$ is a nice well-behaved function. How would I compute the following derivative? $$\frac{\partial}{\partial x} \left\{ \int\limits_0^t \int\limits_{x - t + \eta}^{x + t - \eta} ...
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1answer
16 views

How do i calculate the number of subintervals n in Midpoint method?

I want to calculate the least error (o) in order to obtain the exact answer for integration using the midpoint method. However I am having trouble doing so since i was given a functions whose second ...
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0answers
20 views

Is this identity concerning derivatives true? [duplicate]

Let $f:[a,b]\rightarrow \mathbb{R}$ be a function and $x_0\in \mathbb{R}$. Suppose that $f$ is differentiable in $[a,b]\setminus x_0$ $\lim_{x\rightarrow x_0^+}{f'(x)}$ exists Is it true, then, ...
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1answer
23 views

Calculate the limitation in complex coordinate.

Can we say that this limitation is zero?!and how can we be sure that such limitation leads to a special value although we can't test all pathes through which z leads to a specific value?$\lim_{z\to 0} ...
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4answers
509 views

Is this a correct/good way to think interpret differentials for the beginning calculus student?

I was reading the answers to this question, and I came across the following answer which seems intuitive, but too good to be true: Typically, the $\frac{dy}{dx}$ notation is used to denote the ...
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2answers
49 views

How would you differentiate this? I can't get anywhere

Let's say that $F$ is a nice well-behaved function. How would I compute the following derivative? $\frac{\partial}{\partial t} \left\{ \int_{0}^{t} \int_{x - t + \eta}^{x + t - \eta} F(\xi,\eta) d\xi ...
2
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2answers
95 views

Arcsin domain under differentiation

according to my solutions manual, the derivative of: $$ f(x) = \arcsin \left(\frac{a}{x}\right)$$ is $$f'(x) = \frac{-a}{x\sqrt{x^2-a^2}}$$ however, my work on this problem has found this answer ...
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2answers
160 views

What's exactly the deal with differentials? (Confessions of a desperate calculus student)

So I don't know if I'm the only one to feel this, but ever since I was introduced to Calculus, I've had a slight (if not to say major) aversion to differentials. This sort of "phobia" started from ...
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0answers
31 views

Differentiation of a unit step function? [duplicate]

How come differentiation of a unit step function is Dirac Delta? Can anybody give me a concrete mathematical proof? I only found some intuitive kind of explanation, that at $t = 0$, slope of $u(t)$ ...
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0answers
11 views

Derivative of energy function for anisotropic material

I am trying to derive the anisotropic part of a constitutive model (an energy function) but I'm stuck in the process... Let's define: $\mathbf{F} $ as the deformation gradient (a 3x3 matrix, the ...
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1answer
25 views

What is the term used for the slope of a line compared to the derivative of an equation?

This may seem like a very simple question to seasoned folks out there, but whenever I see a question that states that variable x changes at a rate of some value, I find it hard to decipher it's ...
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1answer
20 views

Find the highest partial derivative between two functions

I have two functions with the following characteristics: $$\text{1)} f(x) \gt g(x), \, \text{for all $x$ in $(0,1]$}$$ $$\text{2)} f(0)=g(0)=0$$ $$\text{3) Both are twice differentiable and concave ...
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1answer
43 views

Related rates: Two planes converging towards a point

An air traffic controller spots two planes at the same altitude converging on a point as they fly at right angles to each other. One plane is 225 miles from the point and is moving at 450 miles per ...
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1answer
16 views

Is the absolute max/min and the local max/min an $x$ or $y$ value?

For example, when we say that $f(c)$ is an absolute maximum value. Are we saying that the $y$ value obtained at the $x$ value $c$ is the absolute max? And therefore, can we say that $c$ is the $x$ ...
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1answer
101 views

Differentiating a definite integral

I am unsure of how to differentiate the follow expression: $$ \frac{d}{dt} \int_\tau^t \Phi(t,\alpha)\cdot A(\alpha)\cdot x(\alpha) d\alpha $$ where $\Phi(t,\alpha)$ and $A(\alpha)$ are matrices.
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2answers
28 views

Proving constants are differentiable using the definition

I'm currently studying differentiability at college, and I can't for the life of me understand how a constant is differentiable. I am using the definition of differentiability (as $h \rightarrow 0$ ...
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3answers
34 views

Expressing as a composite function to begin differentiating with the chain rule

I'm trying to differentiate $y=(x+1)(x+2)^2(x+3)^3$ using the chain rule, but I am having trouble writing it as a composite function. Any help would be great as I can finish the problem on my own once ...
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5answers
555 views

Where did it come from? (derivative of exponential)

We all know this rule: $\text{If } y = a^{f(x)} \text{ then } y' = a^{f(x)} \: f'(x)\ln a$ In my book there is the example: Find $\frac{d}{dx}\left((x^{2} + 1)^{\sin x}\right)$ According to ...
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2answers
27 views

Why We need to derivate by $dy/dx$ and not by $dy/dr$

I need to find values of "r" in the function $y=e^{rx}$ to for the equation $y''- 2y' -3y = 0$ I'm ok to find solution that derivate by "x" but I don't know why I need to derivate by "X" if I ...
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2answers
124 views

Prove that there exists $\delta>0$ such that for all $(x,y)\in S$, if $\|(x,y)\|<\delta$, then $f(x,y)\le f(0,0)$.

I need help with this question: Let $f,g:\mathbb{R}^2\rightarrow\mathbb{R}$ be two $C^2$ functions, and let $S=${$(x,y)\in\mathbb{R}^2:g(x,y)=0$}. Suppose that $g(0,0)=\frac{\partial g}{\partial ...
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0answers
77 views

Study $ h(x)= \sqrt{x^2-1}-x-3$

Let $g$ be the function defined by $$\begin{array}{lrcl} h : & [1;+\infty[ & \longrightarrow & \mathbb{R} \\ & x & \longmapsto & h(x)= \sqrt{x^2-1}-x-3 ...
2
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1answer
18 views

is my answer correct? derivative of logarithmic functions

I want to check my answer, pleas tell me if it's correct or not first problem $y=\left(\log _{\frac{1}{x}}\left(e\right)\right)$ my answer $y=\frac{lne}{ln_{\frac{1}{x}}}$ ...
16
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1answer
323 views

Proof that the range of a map is determined by its behaviour on the boundary.

Let f be a mapping from an open neighbourhood of the 3-dimensional unit ball to the 2-dimensional plane. Suppose that f is smooth (infinitely continuously differentiable on its domain) and regular ...
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2answers
28 views

Related Rates (Cones)

The question reads: A water tank has the shape of a cone with height $2 m$ and radius $\frac{2}{3}$ m. If water is pumped into the tank at a rate of $3 m^3/hr$ how fast is the water level rising at ...
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0answers
26 views

Can we deduce this from partial differential equation?

Let $W(t,x,y)$ be a function that satisfies the partial differential equation $$\frac{\partial W}{\partial t}+ a\frac{\partial^2 W}{\partial x^2} + b \frac{\partial^2W}{\partial y^2} + ...
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0answers
14 views

Finite difference along a line of $xyz$ coordinates

I have a line of $x,y,z$ coordinates. I'm looking at using finite differences in order to find local maxima (of $z$)..i.e. 'hills'. Am I right in thinking that I can use the second derivative? I have ...
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0answers
35 views

If $f$ is differentiable and $f'(a)<0$, is $f$ decreasing on some interval around $a$?

Refer to this question (answer of Rick) :Here The question let $f:[r,s]\longrightarrow \mathbb R$ continuous on $[r,s]$ and differentiable on $(r,s)$. Suppose $f'(a)<0<f'(b)$. Show that there ...
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1answer
35 views

Proove: $\text{arcsinh}(x)>\ln(1+x),\; x>-1$

Proove: $$\text{arcsinh}(x)>\ln(1+x),\; x>-1$$ I tried to prove that $f'(x)>0$ and then $f(x)$ is increasing, but I couldn't conclude it. Is there another way to prove this?
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1answer
50 views

Find $\nabla \|x^HAx - b\|_2^2$

I have to find- $$\nabla\|x^HAx - b\|_2^2,$$ where $x$ is a vector and $A$ is a $4\times 4$ hermitian matrix. I am trying to solve it by using identities given in ...
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1answer
12 views

A calculation method of directional derivative

Let $D_\mathbf vf(\mathbf x) = \lim_{t\to 0} {f(\mathbf x + t \mathbf v) -f(\mathbf x) \over t} $ be the directional derivative of f in the point $\mathbf x$ and direction $\mathbf v$, let $ ...
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1answer
83 views

How to explain aspects of derivatives to little brother?

So as the title suggests, my $12$ year old little brother loves math. Since he is a bright kid, I started teaching him derivatives. The issue is he always keeps asking weird questions that are above ...
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2answers
62 views

Proving that an inequality is true, from assuming that second derivatives exist, and first derivatives are zero on the boundary,

EDIT 2: I just posted my revised proof, where I used two Taylor expansions, and subtracting both equations to get something that's pretty close to what I want. What do you think? Please see below. ...
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0answers
45 views

3 questions over the function $f(x_0,x_1,x_2)=\frac{1}{1+x_0+x_1+x_2}(x_0+x_1,x_1+x_2,x_2+x_0).$

Let $D=\{(x_0,x_1,x_2)\in\mathbb{R}^3|\ 1+x_0+x_1+x_2 =0\}$ the plane in $\mathbb{R}^3$ passing for the points $(-1,0,0)$, $(0,-1,0)$, $(0,0,-1)$ and let $E=\mathbb{R}^3-D$ its complement. Note that ...
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1answer
28 views

Derivation using Leibniz's notation with change of variable

Given the following differential equation: $x^2 \cdot \frac{d^2Y}{dx^2} + p \cdot x \cdot \frac{dY}{dx} + q \cdot Y = 0$ with $p,q \in \mathbb{R}$ I have to prove that by using the change of ...
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0answers
53 views

if f'(a)<0<f'(b) then there is c such that f'(c) = 0 [duplicate]

Let's assume $r < a < b < t$ , and a function $f:[r,t] \to \mathbb{R}$ , $f $ differentiable in $(r,t)$, with $f'(a)< 0 < f'(b)$ . Probe that there exists $c$ such that $a<c<b$ ...
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0answers
46 views

Calculate derivative: $\frac{d^\beta}{d\alpha^\beta}\frac{d^\alpha}{dx^\alpha}\sin(x)$

Is it possible to "calculate" / simplify this expression? If it is, how can it be done? $$ \frac{d^\beta}{d\alpha^\beta}\frac{d^\alpha}{dx^\alpha}\sin(x) $$ for $ \alpha,\beta\in\mathbb{R}_{\ge0} $ ...
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3answers
35 views

Differential of a matrix function

Let $n \geq 1$. Calculate the differential of the function G: $$G: M_n( \mathbb{R}) \times M_n( \mathbb{R}) \rightarrow M_n( \mathbb{R})$$ $$(A,B)\rightarrow A^tBA$$ My first thought was to add ...
1
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2answers
42 views

Natural logarithmic derivative trick

Hi chaps and chapesses, I was wondering if someone could just explain something. If I have a function which is dependent on $x$, the familiar $f(x)$. Now, if I take the derivative of this, and ...