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

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24 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
16 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 ...
3
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1answer
27 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 ...
1
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1answer
34 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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2answers
35 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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2answers
23 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 $\lim\limits_{x\to \infty} f^{'}(x)$ ...
2
votes
2answers
30 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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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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2answers
51 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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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
31 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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2answers
65 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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0answers
38 views

matheproblemask [on hold]

y=logxtopowerxpleasegivemeanswer
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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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0answers
32 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
54 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
18 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 ...
1
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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? ...
3
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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 ...
2
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1answer
33 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 ...
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0answers
21 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 $$ ...
4
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1answer
26 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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1answer
40 views

Real Analysis Question- Differentiability on an interval

So this is the question I am trying to answer... At $(-2,0)$ and $(0,2)$ we just differentiate using normal rules of calculus, yes? Here is my attempt for at $x=0$. Is this correct? For d)I think ...
3
votes
3answers
103 views

Easy question : $\int (xdy+ydx)$

I am ashamed to ask such an easy question but, well: Lets say I got a function $$ f(x,y)=xy $$ Now let's compute the total differential of the function $$ d(f(x,y))=xdy+ydx $$ Now if I do $$ \int ...
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1answer
27 views

Find derivative using the definition of the derivative as a limit

$$f(x) = \frac{1} {\sqrt{x}}$$ find $f'(x)$ using the definition of the derivative as a limit. I know that $$ f'(x) = \frac{(x + \delta)^{-1/2} - (x)^{-1/2}}{\delta} $$ as $\delta$ goes to $0$. ...
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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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7answers
381 views

Understanding derivatives

I don't know if this is written somewhere else. I've looked all over the internet so apologies if this has already been covered. I'm doing Year 12 Maths in Australia for what it's worth. In our ...
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0answers
13 views

selecting points on a domain which represent the derivative of a function

I'm working on some algorithm part of which entails me to subdivide a domain based on the derivative of a function. Let's just consider the 1D case with a closed and bounded domain $[a,b]$ for a ...
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2answers
74 views

derivative of $\ln(4)$

what is the derivative of $\ln(4)$? I am trying to find the derivative of this equation: $h(x)=\ln(\frac{x^3\cdot e^x}{4})$ by rules of logs I simplified the $h(x)$ to the following: ...
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0answers
21 views

Generalized “softmax”

I'm looking for a function $f$ from $\mathbb{R}^n$ to $[0,1]^n$, similar to softmax in the sense that is satisfies these properties: $\sum_i f(x)_i = c$, where $c$ is a chosen constant (i.e., c=1 ...
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2answers
88 views

Limit of $\frac{f\left(x+g\left(x\right)\right)-f\left(g\left(x\right)\right)}{x}$ as $x\to 0$

I'm trying to answer the following question: Let $f$ be continuously differentiable in all of $\mathbb{R}$ and let $g:\mathbb{R}\to\mathbb{R}$ be a function satisfying ...
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2answers
33 views

Using the same limit for a second derivative

I've been trying to answer the same question answered here: Second derivative "formula derivation" And I'm stuck in a step that is not addressed both in the answer and in the comments of ...
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0answers
17 views

diffeomorphism on intervals on R

I came across a line in a proof which involved a diffeomorphism $f:I_1 \rightarrow I_2$ (with $f$ a homeomorphism, $f,f^{-1}\in C^{\infty}$) mapping open intervals in R, which claimed that ...
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1answer
36 views

Solving second order nonhomogeneous linear equation

So i have the equation $$\frac{d^2y}{dt^2} + y = \sin(t)$$ I know the first step is to find the corresponding homogeneous equation, which i think would be: $$r^2+1=0$$ giving real roots and therefore ...
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1answer
24 views

Check differentiablity of $f$ [on hold]

Consider a function \begin{equation*} f(x)=|\cos x|+|\sin (2-x)|. \end{equation*} At which of the following points f is not differentiable? a)$\{(2n+1)\frac\pi2:n\in \Bbb Z\}$ b)$\{n\pi:n\in \Bbb ...
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0answers
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A “prove or disprove question” on absolutely continuous functions [on hold]

Let $f:\left[a,b\right]\rightarrow\mathbb{R}_{+}$ be an absolutely continuous function. ($a<b$). Prove or disprove that the right ( respectively left) derivative of f exists at each point of the ...
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1answer
38 views

Are the extrema of this function global or local?

Last question about this function, I promise. The function $f: \mathbb R \rightarrow \mathbb R$ is given by $$f(x) = \begin{cases} \frac{x^2+5x+7}{x+3} & \mathrm{for} \; x < -3 \\ 0 & ...
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1answer
21 views

Derivatives of real part of function

My physics textbook gives me the complex form equation of simple harmonic motion as: $$z = Ae^{i(\omega _{o}t+\phi )}$$ and then defines $$ x = Re (z) $$ From there they argue that $$ \frac{\partial ...
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1answer
60 views

Chain rule and a function of a function of a function

Suppose we have a composite function: $f(g(h(x)))$, and we want $\frac{\partial f}{\partial h}$. By the chain rule $\frac{\partial f}{\partial h} = \frac{\partial f}{\partial g}\frac{\partial ...
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3answers
51 views

Derivative of a Rational function $f(x)=\sqrt{2x-5\over3x+1}$

I'm trying to find the derivative of, $$f(x)=\sqrt{2x-5\over3x+1}$$ I think I can change this into $$f(x)= \left({2x-5 \over 3x+1}\right)^{1\over2} \\ =[(2x-5)(3x+1)^{-1}]^{1 \over 2}$$ Am I not ...
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1answer
28 views

Deciding whether $(ax+b)/(cx+d)$ is increasing

In order of studying usual function such us : 1) $f(x)=ax^2$ 2) $g(x)=\frac{a}{x}$ 3) $h(x)=\frac{ax+b}{cx+d}$ 4) $p(x)=ax^2+bx+c$ To know if a function is increasing or decreasing we can ...
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1answer
83 views

Feynman Integration Problem

$$ I = \frac{\pi^2}{8} - \int_0^1 \frac{\arctan(x)}{\sqrt{1-x^2}} \,dx $$ Evaluate $I$ $$ I = \frac{\pi^2}{8} - \int_0^1 \frac{\arctan(x)}{\sqrt{1-x^2}} \,dx$$ $$f(a) = \int_0^1 ...
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1answer
35 views

Critical Point of $\mathbb{R}^2 \to \mathbb{R}^2$ function

Given a function $f:\mathbb{R}^2 \to \mathbb{R}$ I can find critical points by finding the $1\times 2$ Jacobian matrix, setting each partial derivative equal to zero and solving the equations. I can ...
8
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1answer
85 views

If $f(0)=f(1)=f(2)=0$, $\forall x, \exists c, f(x)=\frac{1}{6}x(x-1)(x-2)f'''(c)$

Let $f:[0,2]\to \mathbb R$ be a $C^3$ function such that $f(0)=f(1)=f(2)=0$ Prove that $\forall x\in[0,2], \exists c\in[0,2], f(x)=\frac{1}{6}x(x-1)(x-2)f'''(c)$ This problem got me stuck. I ...
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1answer
27 views

Can someone please help me in proving this?

Let $k_{2}>k_{1}>0$, prove that for any $x>0$, $f(x)$ is a monotonically increasing function. $$ f(x)=\frac{1-e^{-k_{1} x}}{1-e^{-k_{2} x}}. $$ We can have ...
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2answers
27 views

With $f(x)= 32 \cosh(x) \sinh(2x) $, determine the slope of its tangent at $( \ln 2 , \, 75)$

With $f(x)= 32 \cosh(x) \sinh(2x)$, determine the slope of its tangent at $( \ln 2 ,\, 75)$. My work $$\sinh x \cosh y = \frac{1}{2}(\sinh (x + y) + \sinh (x - y))$$ $$\cosh(x) \sinh(2x)= ...