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

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3answers
82 views

How to evaluate this exponential fraction limit?

I am trying to determine if 3$^n$ grows faster than 2$^{2n}$. One way I found online to do this was, from Growth was to evaluate $\lim_{n\to \infty} \frac{3^n}{2^{2n}}$ and if that limit evaluates ...
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2answers
38 views

Does dimension of derivative should agree with dimension of variable?

Suppose variable $x$ is a N*1 vector, $A$ is a M*N matrix and $b$ is a M*1 vector. $$ f(x) = \|e^{Ax} - b \|_2^2 $$ Does its derivative should be like following? $$ \frac{\partial f}{\partial x} = ...
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1answer
45 views

Notation for a derivative

I am interested if there is notation for a derivative that is in between a total derivative and partial derivative. The total derivative of $f(t,x,y)$ with respect to $t$ is $$ ...
2
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1answer
86 views

If $f+f'<\varepsilon$, then $f'<\varepsilon$

Let $f$ be continuous on $[a,b]$ and differentiable on $(a,b)$ and there exstis $\varepsilon>0$ such that $f(x)+f'(x)<\varepsilon$ for all $x\in (a,b)$. Prove that $f'(x)<\varepsilon$ for ...
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1answer
31 views

Looking for a standard function $y(x)$ when $dx/dy = ay+b$

I am looking for a standard function $y(x)$, for which the following applies: $dx/dy = ay + b$, with $a$ and $b$ both being constants. Many thanks in advance!!!
2
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0answers
28 views

Proof that derivative operator cannot be written in terms of composition operator (without limits)

Difference delta operator can be written without a limit: $$\Delta[f(x)]=f(x+1)-f(x)$$ The same is true for any other finite difference operator. But what about derivative? Is there a proof that it ...
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1answer
45 views

derivatives of non-analytic smooth functions

I would like to know how to calculate the derivative of a non-analytic smooth function? Suppose $f:\mathbb R\rightarrow \mathbb R$ is in $\mathcal C^\infty\backslash \mathcal C^\omega$ and in ...
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0answers
34 views

acoustic dipole volume integral w/ dirac delta?

I have an acoustic research problem that leads to the following integral formulation: \begin{align} \int_{-\infty}^{\infty}\int_{-\infty}^{\infty}p(\mathbf{y},\tau)\frac{\partial}{\partial ...
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1answer
46 views

Value of x of which a slope is undefined for a parametric graph.

For what values of $x$ is the slope undefined for the graph $$x=8-t^3$$ $$y=t^2-6t$$ The slope should be undefined when $\frac {dx}{dt}=0$. $$\frac {dx}{dt}=-3 t^2$$ $$-3t^2=0$$ $$t=0$$ When ...
3
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1answer
80 views

Continuity of left derivative implies differentiability?

Suppose $f:\mathbb{R}\rightarrow \mathbb{R}$ is continuous and has a left derivative, $f^-$, everywhere in a neighborhood of $x.$ Suppose $f^-$ is continuous at $x.$ Does this imply that $f$ is ...
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0answers
16 views

Is $\int_{0}^{1}\nabla f({t\mathbf{x}})\mathrm{d}t$ ($t\in\mathbb{R}$, $\mathbf{x}\in\mathbb{R}^n$) differentiable with respect to $\mathrm{x}$?

Consider a differentiable vector function $f:\mathbb{R}^n\longrightarrow\mathbb{R}$. Is the single-variable integral of its gradient $$\int_{0}^{1}\nabla f({t\,\mathbf{x}})\,\mathrm{d}t\qquad ...
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1answer
23 views

Finding derivatives for a Cauchy-Euler ODE

I'm having some trouble following along with the reduction of the Cauchy-Euler equation into a linear one with constant coefficients. I've been trying to follow along with the work here, but I don't ...
1
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1answer
44 views

Trick with differentials from $\frac{dr}{ds} \to \frac{dr}{dt}$

I need to come from $r''=\frac{d^2r}{ds^2}$ to $\ddot{r}=\frac{d^2r}{dt^2}.$ I know that $r' = \frac{\dot{r}}{|\dot{r}|}.$ And more $1/\dot{s} = t' = \frac{1}{\sqrt{\dot{r}\cdot\dot{r}}}.$ You can see ...
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2answers
43 views

What are the critical points of $x-4\sqrt{x+1}$?

A critical point $c$ is defined as $f'(c) = 0$ or $f'(c) = $ undefined. This definition is taken from this video. if $$f(x) = x-4\sqrt{x+1}$$ then $$f'(x) = 1 - \frac{2}{\sqrt{x+1}}$$ To find the ...
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2answers
74 views

Prove that $\nabla_X tr(X^TAX)= (A+A^T)X $ where $A \in \mathbb{C^{m*m}}$ and $X \in \mathbb{R^{m*n}}$

Prove that $\nabla_X tr(X^TAX)= (A+A^T)X $ where $A \in \mathbb{C^{m*m}}$ and $X \in \mathbb{R^{m*n}} $ . 1.) Same proof stands when $ A\in \mathbb{C}$ or $ A\in \mathbb{R}$ ? 2.) What is the ...
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2answers
72 views

How do you find the inflection point of this graph?

The graph is this: $$ \frac{(x+1)^3 - 4(x+1)^2 + 4(x+1)}{(x+1)^2 - 2(x+1) + 1} $$ I know you can find second derivative and then solve for values that make it undefined or 0, but I was told ...
2
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1answer
73 views

When $ \lim_{x\rightarrow +\infty}\frac{f(x)}{x}=0$ implies $\lim_{x\rightarrow +\infty}f'(x)=0$?

I have just solved a problem: Let $f:[0,+\infty)\rightarrow \mathbb{R}$ be continuous on $[0,+\infty)$ and differentiable on $(0,+\infty)$. If $\displaystyle \lim_{x\rightarrow +\infty}f'(x)=0$, ...
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1answer
54 views

Proving question (Differentiation)

Given that $y=\tan (x+\frac{\pi}{4})+1$, show that $\frac{d^2y}{dx^2}=2y(\frac{dy}{dx}$). I got $\frac{dy}{dx}=\sec^2 (\frac{\pi}{4}+x)$ but I don't know how to proceed. P/s: The question states ...
2
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1answer
48 views

Prove $f'(0)=0$, if $|f(x)|≤x^2$

Let $f:(-a,a) \longrightarrow R$, $a>0$. Such that $$ |f(x)|≤x^2 $$ What I did was taking out the module bars so I get $-x^2≤f(x)≤x^2$ and I see that at $x=0$ the function must be zero. I see ...
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1answer
28 views

how does this given condition translate into this specific interval (calculus)

in my calculus book i am learning how derivatives affect the shape of the graph, for one problem i am supposed to sketch a graph given the conditions. I understand when f'(x)>0 the f(x) will increase ...
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2answers
95 views

Statement about Rolle's Theorem (true or false?)

There's a statement, that I believe is false Between two distinct zeroes of a polynomial $p$, there is a number $c$ such that $p′(c) = 0$. Here is my reasoning: A polynomial of an even ...
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4answers
132 views

Does $f(x)>g(x)$ imply $\frac{d}{dx}f(x)>\frac{d}{dx}g(x)$?

Is it true that $f(x)>g(x) \implies \frac{d}{dx}f(x)>\frac{d}{dx}g(x)$? What about $|f(x)|>|g(x)| \implies \frac{d}{dx}|f(x)|>\frac{d}{dx}|g(x)|$?
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1answer
48 views

Maximizing trapezoid area

Find the trapezoid of largest area that can be inscribed in the region bound by the graph of $y=4-x^2$, and the $x$-axis. So, I know that you need to maximize the area of the trapezoid, which I used ...
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2answers
45 views

What does the notation $\frac{\partial(x,y)}{\partial(u,v)}$ mean?

Suppose G$(u,v) = (x, y, z)$ In terms of derivatives, what does $\frac{\partial(x,y)}{\partial(u,v)}$ mean?
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2answers
57 views

How to differentiate $y=(x+1)^3/x^{3/2}$ and $y=2x^4/(b^2-x^2)$

I need to solve a list of derivatives to help me on an exam; however, I'm in doubt when they use another variable (constant) or when I have a fraction with functions that use the power rule. For ...
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6answers
305 views

Derivative: $e^x$. [duplicate]

How do you differentiate $e^x$? I looked on many sites, including similar questions here but most answers seemed circular. The only known definition of $e$ to be used in this proof is $$ e=\lim_{n ...
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0answers
35 views

Partial derivatives generalization

Let $F:U\rightarrow\mathbb{R}^m$ be a function for some open $U\subseteq\mathbb{R}^n$. If all partial derivatives of $F$ exist and are continuous in a neighborhood of some point $x_0\in U$, then $F$ ...
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0answers
29 views

Curious coincidence: $ 2x^2 (x^2-1) =3(y^2-1) $ and $ x(x-1)/2 =2^n -1 $

Why is it such a coincidence that for both diophantine equations $ 2x^2 (x^2-1) =3(y^2-1) $ and $ x(x-1)/2 =2^n -1 $ have five positive integer solutions $x=1,2,3,6,91$? Source: example 29
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3answers
1k views

Computing a higher (2015) order partial derivative of $1/(x^2+y^2)$

Suppose $$f(x,y) = \frac{1}{x^2 + y^2}\text{.}$$ Find $$\frac{\partial^{2015} f}{\partial x^{2015}}\text{.}$$
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1answer
44 views

Differentiation / Limit problem from intro real analysis textbook

Assume that $f: (-1,1) \to \mathbb{R}$ and $f'(0)$ exists. If the sequences $\alpha_n, \beta_n \to 0$ as $n \to \infty$, define the difference quotient $$D_n = \frac{f(\beta_n) - f(\alpha_n)}{\beta_n ...
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2answers
48 views

Why only the numerator is derived?

Why the derivative of $y = \frac{x^5}{a+b}-\frac{x^2}{a-b}-x$ is solved by deriving just the numerators? The solution is $\frac{dy}{dx}=\frac{5x^4}{a-b}-\frac{2x}{a-b}-1$.
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1answer
37 views

How to take derivative of algebraic function with respect to s

I am reading some books and papers on operational calculus (which is quite similar to Laplace transform) and am unable to understand some of the workings/derivations. So I guess I need to understand ...
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2answers
28 views

Prove that $f$ has an inflection point at zero if $f$ is a function that satisfies a given set of hypotheses

Prove that if $f$ is a function such that $f'(x) > 0$ $\forall x \neq 0 \wedge f'(0) = 0 \wedge f''$ is a continuous one to one function on some open interval $(a, b): a < 0 < b$ then $f$ ...
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0answers
19 views

$F(x,y) = (x^2 + 2y^2, 2x^2 +y^2)$ and $A= \{(x,y) : x>0, y>0\}$

$F(x,y) = (x^2 + 2y^2, 2x^2 +y^2)$ and $A= \{(x,y) : x>0, y>0\}$ I need to find the following: $(a)$ Show $F$ is one-to-one on $A$. $(b)$ Show that $F(A) = \{(u,v) : 0 < \frac{u}{2} < v ...
3
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1answer
114 views

How to show that $\frac{-1}{x^2}=0$ has no solutions?

I must check if the function $f(x)=\frac{1}{x}$ has a tangent line with slope $0$. I took the derivative: $$\left[\frac{1}{x}\right]'=\frac{-1}{x^2}$$ And then: $$\frac{-1}{x^2}=0$$ ...
3
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0answers
50 views

How can we show that the functions are differentiable?

Show that the following functions $$f(x, y)=\frac{xy}{\sqrt{x^2+y^2}} \\ f(x, y)=\frac{x^2y}{x^4+y^2}$$ are differentiable at each point of the domain. Determine which of them is $C^1$. $$$$ The ...
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1answer
30 views

Find sum of arguments where function takes supremum and infimum

Let $f(x)=(\tan x)^{\sin 2x} $ $\\$ for $x\in(0, \frac{\pi}{2})$ let $i$ be an argument where function takes infimum and $s$ - supremum. Find $i+s$ I calculated $f'(x)=2e^{\sin2x\cdot\ln{\tan ...
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1answer
42 views

Interchanging differentiation and evaluation

Suppose $f(x,y)\in \mathcal{C}^2$ (twice continuously differentiable, also real). When can you say $$f_x(x,y)\Biggr|_{y=0} = \partial_x\left(f(x,0)\right)$$ with the loosest possible restrictions? ...
3
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2answers
87 views

Understanding higher dimensional derivatives

I'm having trouble understanding higher dimensional derivatives. Suppose $f: \Bbb R \to \Bbb R$. We say $f$ is differentiable at $x = c$ if $\lim \limits_{x \to c} \dfrac{f(x) - f(c)}{x - c}$ ...
3
votes
3answers
168 views

Can the third derivative tell me anything about the original function?

The first derivative can tell me about the intervals of increase/decrease for $f(x)$. The second derivative can tell me about the concavity of $f(x)$. So can the third derivatives, and any ...
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2answers
32 views

What does $\frac{3x^2 + 2ax + b}{2y} \quad$ mean w.r.t the derivative of $y^2 = x^3 + ax^2 + bx$?

We know that if the given equation were $\quad y = x^3 + ax^2 + bx$, $\quad$ then the derivative would be $3x^2 + 2ax + b$. Since the given equation is different so the derivative will be: $$2(x^3 + ...
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2answers
95 views

What is the derivative of a matrix w.r.t itself?

what is the derivative of \begin{equation}\partial \frac{x^TVx}{\partial V} \end{equation} where V is a matrix and x is a vector. In general what is the right way to calculate matrix derivatives w.r.t ...
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1answer
89 views

Proving the Implicit function theorem in a particular case.

Let $g:\Bbb{R^2}\rightarrow\Bbb{R}$ differentiable. Assume that $g(0,0)=0$ and $g'_y$ continuous at $(0,0)$ and $g'_y(0,0)>0$. It is asking to prove the Implicit function theorem in this ...
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2answers
35 views

The equation of the normal to the curve.

The equation of the normal to the curve: $f(x)=x^2-5$ at $x=2$ I know that $y-y_1=m(x-x_1)$ But I don't really know how to procede
2
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1answer
84 views

How does this “integration by differentiation” method work

Apparently, the integral of a function f(x) from a to b can be done through differentiation through this method: $$ \int_a^b f(x)dx = \lim_{x \rightarrow \ 0 } f(\frac{d}{d x} ...
1
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2answers
58 views

Proof that the derivative of a function $f$ and $g$ are equivalent $\forall x \in$ the domain of $f(x)$ and $g(x)$

Set $ g(x) = \left\{ \begin{array}{lr} \frac{1}{x} & : x > 0 \\ \frac{1}{x} + 1 & : x < 0 ...
1
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0answers
32 views

Question about derivative notation

So i am studying for my calc test and i have a quick question does $dy/dx$ means $y'(x)$? and does $dy/dt$ means $y'(t)$? Thanks
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1answer
42 views

what is the name of the formula $x_{n+1} = x_n - \frac{f(x_n)}{f'(x_n)}$, for every $n \geq 0$

I am solving for $x^x = 100$, I found a solution that used $$x_{n+1} = x_n - \frac{f(x_n)}{f'(x_n)}$$ for every $n \geq 0$ as the working equation in finding the value of $x$. I want to know what ...
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votes
4answers
48 views

Derivative of $ \frac {\sqrt{x^2+1}}{2}$

So the wolfram alpha says the answer is: $ \frac {x}{2 \sqrt{x^2+1}}$ But when I try to take the derivative of: $ \frac {\sqrt{x^2+1}}{2}$ by quotient rule, I get: $ \frac {\frac{1}{2} \cdot ...
0
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1answer
44 views

Can't solve second order ODE with variation of parameters or undetermined coefficients

I have to solve $$y''+4y' +y=\frac{e^\left(-2x\right)}{x^2}$$ The homogenous equation is easy enough to solve and I got $$y(x) = c_1e^{-2 + \sqrt{3}} + c_2e^{-2 - \sqrt{3}}$$ Doing variation of ...