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

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Circular definition of tangent line and derivative

I'm trying to understand the deep relations between the tangent line to the graph of a function $f$ at a given point $P$, and the derivative of $f$ at the same point. Indeed, in many books the ...
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1answer
54 views

Polynomial satisfying $f(x) = f'(x) \cdot f''(x)$

If a polynomial of degree $n$ satisfies $f(x) = f'(x)\cdot f''(x)$ (such that $n$ belongs to $\mathbb R$) then $f(x)$ is? A) an onto function B) an into function C) no such function possible D) ...
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0answers
25 views

prove: $\triangledown(\triangledown\cdot u)-\triangledown \times (\triangledown \times u) =\vartriangle u$ [duplicate]

The claim is $\triangledown(\triangledown\cdot u)-\triangledown \times (\triangledown \times u) =\vartriangle u$ where $u:\mathbb{R}^3\to \mathbb{R}^3$ is a vector field, $ \vartriangle$ is the ...
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1answer
36 views

Directional derivative $f(x,y)=\frac{x^3}{1+x^2+y^2}$

I'm stuck on calculating the directional derivative of $f(x,y)=\frac{x^3}{1+x^2+y^2}$ in $(3,-1)$ along $(a,b)\in\mathbb{R}^2$. My try: $\lim\limits_{t\to ...
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3answers
46 views

The derivative of $x!$ and its continuity

is the factorial of fractions and negative numbers defined? If yes, then what is its graph? Also please find its domain. Our teacher said the factorial of a fraction is the fraction itself. He also ...
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1answer
19 views

Rate of change of a multivariable equation w/ respect to another equation

So I was told to find the rate of change of $$ f(x,y) = x^2 − 3xy + y2 $$ with respect to $$r(t) = e^{2t}+t^2$$ I know usually I would take the derivative with respect to each variable and then ...
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1answer
31 views

Derivative of $|x|$

If $f(x)=|x|$ for $x = (x_1,x_2,x_3,\ldots,x_n) \in \mathbb{R}^n$, what is the derivative of $f(x)$ with respect to $x_i$ if $i\in\{1,2,3,\ldots,n\}$? I am confused, please show me a hint to get ...
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1answer
14 views

Proving a corollary of a corollary of the Mean Value Theorem (corollary-ception)

This is will a wordy question but here it goes: My analysis book states the mean-value theorem and then a corollary which we will label as (1): Let $f$ be a differentiable function on $(a,b)$ such ...
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2answers
55 views

One point following another moving in a straight line?

There is a plane with two points on it, let's say A and B. A starts at an arbitrary constant point, let's say $(0, 0)$, and $B$ at a point that needs to be tested, which we'll call $(c, d)$. A moves ...
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1answer
46 views

A function with midpoint-linear derivative is a quadratic polynomial

Suppose $f:\mathbb{R}\to \mathbb{R}$ is a differentiable function such that $$f'\left(\frac{a+b}{2}\right) = \frac{f'(a)+f'(b)}2,\quad \forall a,b\in\mathbb{R}$$ Prove that $f$ is a polynomial of ...
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3answers
123 views

Functions $f$ such that $f(x+1)-f(x-1)=2f'(x)$.

What can one say about functions $f:\mathbb{R}\to\mathbb{R}$ satisfying the condition $f(x+1)-f(x-1)=2f'(x)$? Is is possible to find all such functions, or is this defining equation the best ...
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2answers
45 views

Is there has a smart way to compute the 1 order derivative of the circle equation? [closed]

I have encountered a compute problem. This exercise has given the circle equation and a para-curve equation with unknown parameters, the para-curve and circle has the same radius of curvature, and ...
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2answers
35 views

gradient norm of a simple function

In this answer Derivation of soft thresholding operator how can I derive that $\nabla(||x-b||_2^2)=b-x$?
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1answer
55 views

What are higher derivatives?

From Wikipedia: Higher derivatives can also be defined for functions of several variables, studied in multivariable calculus. In this case, instead of repeatedly applying the derivative, one ...
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1answer
20 views

Existence of a smooth function with given derivative roots

Is there a smooth function $f$ that for all $n\in\mathbb{Z}_+$, $f^{(n)}(n)=0$ i.e. $n$th derivative at the point $n$ is zero and $f^{(n)}(x)\ne 0$ for all $x\in\mathbb R\setminus \{n\}$? If there is ...
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1answer
32 views

Extreme values of a two-variable polynomial

Is it possible to find a two-variable polynomial which has only two extreme values on the whole plane, one is a local maximum, another is a local minimum, and the local maximum is less than the local ...
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1answer
38 views

Derivative of the composition of two functions

Is the calculation below valid? \begin{align} f(x)=ax+b+g(f(x))\\ \frac{df(x)}{dx}=a+\frac{dg(f(x))}{df(x)}\frac{df(x)}{dx}\\ \frac{df(x)}{dx}-\frac{dg(f(x))}{df(x)}\frac{df(x)}{dx}=a\\ ...
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1answer
38 views

Different results on doing $\frac{\partial}{\partial y}\left(\int_r^y \frac{1}{\sqrt{y^2-s^2}} ds \right)$ in different ways

I have a confusion when trying to get the result of the expression below, $$ I = \frac{\partial}{\partial y}\left(\int_r^y \frac{1}{\sqrt{y^2-s^2}} ds \right). $$ All variables are real and $y>r$. ...
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2answers
26 views

How to evaluate limits

Let $f$ be a continuously differentiable function on $\mathbb R$. Suppose that $L=\lim\limits_{x\to \infty}(f(x)+f^{'}(x))$ exists. If $0<L<\infty$, and if $\lim\limits_{x\to \infty} f^{'}(x)$ ...
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2answers
31 views

Differentiating both sides of an inequality with monotonic functions

If $f(x)\le g(x)$ for all real $x$ for monotonic functions $f$ and $g$ (say, both increasing), does it follow that $f'(x)\le g'(x)$? (Note: I've seen several questions asking the same thing without ...
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2answers
252 views

$\alpha$-derivative (concept)

I found the following definition: Given an real number $\alpha$, we say that a function $f: \mathbb{R} \rightarrow \mathbb{R}$ is $\alpha$-differentiable at $0$ if exists the limit: $$\lim_{t \to 0^+} ...
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0answers
16 views

Differential calculation in multiple variables function

This question is somehow related to this question. Consider a multiple variables function $G(u, v) = \left(\matrix{x(u,v) &=& G_x(u,v)\\y(u,v) &=& G_y(u,v)\\z(u,v) &=& ...
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4answers
14k views

Construct a function which is continuous in $[1,5]$ but not differentiable at $2, 3, 4$

Construct a function which is continuous in $[1,5]$ but not differentiable at $2, 3, 4$. This question is just after the definition of differentiation and the theorem that if $f$ is finitely ...
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2answers
71 views

A question about the vector space spanned by shifts of a given function

Let $E$ be the space of continuous real functions and $f\in E$ Let $T_t$ denote the shift operator: $T_t(f)(x)=f(t+x)$ Let $T(f)$ be the linear span of the set $\{T_t(f) \;|\; t\in ...
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1answer
25 views

Converting prime notation of derivatives to Leibniz notation.Resources needed

I have been studying calculus for past few months and through the time I have been using the so called prime notation.I have been studying from Spivaks Calculus for those of you who are familiar with ...
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1answer
54 views

Antiderivative of $\frac {dy}{dx}$

This is probably a very simple question, but I think its interesting. What I would think, based on my intuition (which I think is correct in this case) is that $$\int \frac {dy}{dx}=y$$ However, ...
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2answers
108k views

Calculus question taking derivative to find horizontal tangent line

How would I solve this problem? Find the point where the tangent line is horizontal in the following function: $$f(x)=(x-2)(x^2-x-11)$$ I computed the derivative: $\quad ...
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5answers
14k views

What exactly is the difference between a derivative and a total derivative?

I am not too grounded in differentiation but today, I was posed with a supposedly easy question $w = f(x,y) = x^2 + y^2$ where $x = r\sin\theta $ and $y = r\cos\theta$ requiring the solution to ...
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1answer
12 views

Is the following property suffictient for second order differentialbility?

Let $U\subset R^n$ be an open set, and $f:U\to\mathbb R$ a $C^1$ function. Suppose that for any $x_0\in U$, there exists a $n$-variable-polynomial $T_{x_0}$ of degree at most $2$ such that, ...
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0answers
32 views

upper bound of an $L^\infty$ function's derivative

Consider a function $u:\mathbb{R} \longrightarrow \mathbb{R}^n$ that is essentially bounded, i.e., $u \in L^\infty$. There is an upper bound of its derivative? I think there is not allways ( i.g. ...
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1answer
37 views

Apply chain rule to $u = y^{1 - n}$ in order to find $\frac{du}{dx}$

Let $u = y^{1 - n}$. I know that, by using the chain rule: $$\frac{du}{dx} = \frac{du}{dy} \cdot \frac{dy}{dx}$$ Also, I know that $\frac{du}{dy} = (1 - n)y^{-n} = \frac{1 - n}{y^{n}}$ Now, for ...
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1answer
42 views

Why is the chain rule applied to derivatives of trigonometric functions?

I'm having trouble to understand why is the Chain rule applied to trigonometric functions, like: $$\frac{d}{dx}\cos 2x=[2x]'*[\cos 2x]'=-2 \sin 2x$$ Why isn't it like in other variable derivatives? ...
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33 views

How fast is the distance between two points changing.

I am having a difficulty with the following question from my calculus unit. Bus station A is located 100km west of bus station B. At 12pm a bus leaves station A driving south at 70km/h and a bus ...
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1answer
56 views

$f '(x) = -f(x)$ and $f(1) = 1$, Solve for $f(2)$

I am honestly not even sure how to start this problem... My first thought was that $f(2) = 2$ ... But now I don't even know where to go from there.
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1answer
20 views

Find the line normal to the curve at a specified point [on hold]

Find the line normal to the curve $xy^2 + 2xy = 8$ at the point $(1,2)$.
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0answers
12 views

Mean Value Inequalities for vector-valued functions

Let $X$ and $Y$ be Banach spaces, and let $U\subset X$ be open. If $f\colon U\to X$ is continuously differentiable and $x,v\in X$ are such that the line segment $\ell=\{x+tv\mid t\in[0,1]\}$ lies ...
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1answer
97 views

how to solve this limit with $e^{x}$

I was trying to solve the derivative of $e^{x}$ the traditional way with the definition of the derivative: $$ \lim_{h\rightarrow 0}\frac{e^{x+h}-e^{x}}{h} $$ so I solved like this: ...
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2answers
59 views

Cube Root function not differentiable

Why is the cube root function not differentiable at x=0? I graphed it and the curve looks a bit vertical at that point, is that why? Can someone give a good explanation please.
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1answer
32 views

Stochastic calculus - Ito confusion

We have $W(t) = f(t)X(t)$. My textbook says that $dW = fdX + X\dfrac{df}{dt} dt$. I don't get how they arrived at this conclusion. I get the first part, because $\dfrac{dW}{dX}dX = fdX$. But for the ...
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1answer
526 views

The normal line intersects a curve at two points. What is the other point?

The line that is normal to the curve $\displaystyle x^2 + xy - 2y^2 = 0 $ at $\displaystyle (4,4)$ intersects the curve at what other point? I can not find an example of how to do this equation. Can ...
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1answer
34 views

Understanding the Definition of a derivative as slope of a tangent line

I'm trying to understand the derivative and am wondering why the derivative is described as the slope of the tangent line and not the slope of a function itself. Say $f(x) = 2x+5$ where ...
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1answer
10 views

What is the mean value theorem for the Fréchet (total) derivative?

What is the mean value theorem for the Fréchet (total) derivative? Off the top of my head, it's something like $$ \|F(x+h)-F(x)\|\leq \sup_{c\in[0,1]} \|F^\prime(x+ch)\|\|h\| $$ but the double ...
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3answers
47 views

Maximum & minimum area of rectangle outside another.

Find the maximum & minimum area of an outer rotated rectangle when the inner rectangle has the side lengths $a$ and $b$. Here's an image: I have already tried to relate the side of ...
2
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1answer
1k views

Minimizing L1 Regularization

I have given a high dimensional input $x \in \mathbb{R}^m$ where $m$ is a big number. Linear regression can be applied, but in generel it is expected, that a lot of these dimensions are actually ...
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1answer
27 views

Can I interpret the exponential of the derivative operator, $e^D$, as infinite shift operators each shifting “infinitesimally”?

To better explain what I mean, an example can be very useful. Consider $e^{i\theta}$. We could express this using the series definition or the limit definition of $e^x$ instead: $$e^{i\theta} = ...
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0answers
22 views

Grand canonical derivative.

I've been trying to work out how to find the density in the thermodynamic limit of a nearest neighbour magnetic lattice gas in the grand canonical ensemble. I'll with hold the Hamiltonian for the ...
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0answers
8 views

When can we move a Fréchet derivative under a Lebesgue integral?

Under what conditions can we move a Fréchet derivative under a Lebesgue integral? Specifically, when does $$ G'(x) = h\in X\mapsto \int_{\Omega} \left(F_x^\prime(x,t)h\right) \mu(dt) $$ where $$ ...
2
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4answers
58 views

Real Analysis - differentiable

$f:[0,\infty]\rightarrow \mathbb{R}$ is twice differentiable. If $f''$ is bounded and exists the limit of $f(x)$ at infinity, then $\lim_{x\rightarrow \infty}f'(x)=0$. I tried to use the Taylor's ...
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3answers
76 views

Find the Derivative of $\frac{1}{\cos^2(2x)+\sin^2(2x)}$ [closed]

Calculate the derivative of: $$\frac{1}{\cos^2(2x)+\sin^2(2x)}.$$ How would I calculate such a derivative?