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

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Higher Order Derivative Tests in Multiple Dimensions

To evaluate the minima, maxima, and saddle points of a real function of 2 variables, we use the second derivative test after evaluating the critical points to identify the type of extrema they are. ...
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24 views

Forms of functions in dynamical systems

I wanted to read some introductory material about dynamical systems since I might need a basic understanding of them in a related task. So, as far as I see, in a continuous time dynamical system, we ...
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32 views

What are derivatives? [closed]

I know this may be a d/umb question but I really want to know. I am new to calculus i haven't taken it in school yet i just like studying math. I don't know anything about derivatives so anything you ...
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15 views

Proving non-differentiability in a basic sense

I've tried to make this question general enough that it solves other users' questions! Perhaps a solution to this post will explain how one might use a basic definition (see $\mathbb{\S}$) to prove ...
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16 views

Question about linearising system with second derivative

I need to linearise a system: $\ddot{x}+4\dot{x}^5+(x^2+1)u=0$. The referenced answer is :$\ddot{x}+0+(0+1)u\approx0$. So, the linearly approximated about $x=0$ is: $\ddot{x}=-u$ I can understand ...
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36 views

Do irrational derivative orders exist?

There are many notations for a derivative of $y$ with respect to $x$. Two, most popular are $y'(x)$ or just $y'$ and $\frac{dy}{dx}$. For higher order derivatives, the more consistent notation is ...
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56 views

Difference in use between $d$, $\partial$, $\operatorname d$, $\varDelta$ and $D$ for derivatives.

While reading different sources on implicit differentiation (and thereafter differentiation in general), I came across many different "d's" being used for (or similar to) the familiar ...
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29 views

Matrix Calculus and Linear Transformations

I'm working on making the jump from differentiating real valued functions ($f: \mathbb{R}^n \rightarrow \mathbb{R}$) and vector valued functions ($g: \mathbb{R}^n \rightarrow \mathbb{R}^m$) to matrix ...
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59 views

Prove that $\exists x_0, x_1\in (0,1)$, such that $\frac{f'(x_0)}{x_0}+\frac{f'(x_1)}{x_1^2}=5$

Let $f:[0,1]\to\mathbb{R}$ be a differentiable function, such that $f(0)=0$ and $f(1)=1$. Prove that there exist different $x_0, x_1\in (0,1)$, such that ...
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19 views

How would I calculate an derivative with two unknown variables?

I'm learning calculus II. I recently wondered what if I had two unknown variables in an function, and wanted to take an derivative. Let's say there is a function $f(x,y)=2x^3+7y^2$ How would I ...
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228 views
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Horizontal tangent line of a parametric curve

Suppose $x=t^2,y=t^3$ is a parametric curve. Here's a quote from my textbook: The origin, which corresponds to $t=0$, is a singular point of the parametric curve, because $dx/dt=2t,dy/dt=3t^2$ are ...
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Differentiating a squared quantity

I was reading through my electromagnetism book where i came across this statement where when we differentiate wrt a squared quantity rather than a single quantity we multiply it by $\frac{1}{2}$. ...
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23 views

Prove that $f$ is analytic and derivatives of all order $f_n^{(k)}$ converge to $f^{ (k)}$ uniformly on any compact subset of $G$.

Suppose $f_n$ is analytic in some region $G$ and suppose $f_n$ converges to $f$ uniformly on any compact subset of $G$. Prove that $f$ is analytic and derivatives of all order $f_n^{(k)}$ converge to ...
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1answer
24 views

Range of this double trigonometric function

What is the range of $$ \sin(\cos(x)). $$ Generally we have various methods. Like differentiation, graphical analysis. So it's range is $[0,1]$. But how to prove it using any known method or ...
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3answers
57 views

Prove that inequality is true for x>0: $(e^x-1)\ln(1+x) > x^2$

I was given a task to prove that inequality is true for x>0: $(e^x-1)\ln(1+x) > x^2$. I've tried to use derivatives to show that the $f(x) = (e^x-1)\ln(1+x)-x^2$ is greater than zero, but has never ...
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35 views

Number of real roots of $f ' ( x )$

Let $$f(x)=(x-a)(x-b)^3(x-c)^5(x-d)^7 $$ where $a,b,c,d$ are real numbers with $a < b < c < d$ . Thus $ f ( x )$ has $16$ real roots counting multiplicities and among them $4$ are ...
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22 views

Calculatin a partial derivative

If we had: and we were to calculate How is it equal to w ?
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32 views

Find $f(1)$ and $f'(1)$ of $\lim_{h\rightarrow\ 0}\frac{f(1+h)}{h} = 5$

Suppose the function, $f$, is differentiable at $x = 1$. $$\lim_{h\rightarrow\ 0}\frac{f(1+h)}{h} = 5$$ Find a) $f(1)$ and, b) $f'(1)$. I know b) (well at least I think it can) can be found by the ...
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7 views

Question conserning the existence and continuity of derivatives of function's shperical mean

I heard a rumor that the claim beneath is true and I'm trying to prove it (or find a counterexample). Let $f:\mathbb{R}^n\rightarrow \mathbb{R}$, $f\in C^k(\mathbb{R}^n)$. Fix $\varepsilon > 0$ ...
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54 views

Prove that $f(x)=\sum_{n=1}^{\infty} (\frac{x}{n}-log(1+\frac{x}{n}))$ is continuous and can be differentiated ad infinitum

We have $f:(0,\infty) \rightarrow \mathbb{R}$ defined by infinite series $f(x)=\sum_{n=1}^{\infty} (\frac{x}{n}-log(1+\frac{x}{n}))$ Prove that $f$ is continuous and can be differentiated ...
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1answer
26 views

Let B be set of all twice differentialbe function $ f(0)=1, f'(0)=-1$ . .. Find supremum of $ {(f''(0):f\in B})$

Let B be set of all twice differentiable function $f$ such that $f: (-1,1) \to (0,\infty)$ and $ f(0)=1, f'(0)=-1$ . We have new function $g(x)$ such that $g(x)=\frac{1}{f(x)}$ and $g(x)$ is convex ...
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derivative of differentiable function [duplicate]

Edited: It is known that if $f$ is differentiable then the derivative function of $f$ is not always continuous. For instance $f(x)=x^2\sin (\frac{1}{x})$ for $x\neq 0$ and $f(0)=0$ if $x=0$. Then ...
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60 views

What does $f'(xy)$ mean?

I apologize in advance for the silliness of such question, but what is the meaning of $f'(xy)$ in $yf'(xy) = f'(x)$? Is it the total derivative of $f$ w.r.t $x$? Or it is the derivative w.r.t $xy$?
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$f \in C^1[0,\infty)$ such that $\lim_{x \to \infty} \dfrac {xf(x)}{f'(x)}=2$ ; then for $s<2$ ; $\lim_{x \to \infty}x^{-s}f(x)=\infty$?

Let $f \in C^1[0,\infty)$ be such that $\lim_{x \to \infty} \dfrac {xf(x)}{f'(x)}=2$ ; then is it true that for $s<2$ , $x^{-s}f(x) \to \infty$ as $x \to \infty$ ?
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Rewriting of functional equation $f(xy)=f(x)+f(y)$

Given the following equation: $f(xy)=f(x)+f(y)$ and the fact that $y=x^{-1}$, I've to find how this could became: $f'(x) = f'(1)/x$, where it is said that $f'(x)$ is the total derivative of $f$ ...
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332 views
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Show that $\dfrac{\rm{d}^{L-m}}{\rm{d}x^{L-m}}\left(x^2-1\right)^L=\dfrac{(L-m)!}{(L+m)!}(x^2-1)^m\dfrac{\rm{d}^{L+m}}{\rm{d}x^{L+m}}(x^2-1)^L$

The question that follows is needed as part of a derivation of the Associated Legendre Functions Normalization Formula: ...
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Different results in differentiation; I don't see the flaw

Might be getting sleepy here, but I cannot find an issue in both methods which yields different results. The function in concern is $f(\theta)=nk \text{log}\theta - (k+1)\sum^n ...
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1answer
45 views

Calculus rates of change -modified version of a classic problem

I have the following problem: Suppose there is a car moving at constant speed of 100 mi/h from left to right following a path described through the function y=x^2. Also, there is another car moving ...
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1answer
33 views

Second distributional derivative of cosine

I need to compute second distributional derivative of the function $$ g(x) = cos|x-2|, $$ but I'm not sure about my solution. \begin{align} \left<g'', \varphi \right> = \left<g, ...
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How do you find the PDF when you are given the new variable wrp to a known random variable?

My question is rather simple but here's a specific example I'd like to work with. The pareto distribution is given by the PDF $f(y:\theta)=\theta y^{-\theta-1}$ and $y_i$ are distributed with this ...
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2answers
57 views

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

How does this $\frac{x+\sqrt{x}}{x^2}$ to $x^{-1} -\frac{3}{2}x^{-\frac{5}{2}}$ ? I don't need the final answer just everything in between. I'm not sure what rules let to this. Here is an image, I ...
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62 views

How does differentiation work?

I am a physics student and my teacher told me, to find the instantaneous velocity of an object, reduce the time interval to a very small extent. May the time interval be very very very close to 0, ...
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23 views

Adjoint Operator to the Derivative

Let $V \subset \Bbb R[X]$ be the Vectorspace of all Polynomials of degree $\le 3$. The inner product on $V$ is defined as follows: $$\langle f,g \rangle:=\int^1_{-1}f(t)g(t)dt$$Let $L:V \to V$ be the ...
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Maxima/Minima question seems contradictory

Sorry for putting in the picture.I tried but I wasn't able to input the inverse function using Latex. So my question is as given in no. 21. It states that, the function is minimum at $\ x=1$.This ...
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1answer
56 views

Finding a function based on its Derivative without Integrating

My question revolves around finding a function based on its derivative of the type below : Problem : The limit below represents the derivative of some real-valued function $f$ at some real-number ...
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Can we derive a Taylor formula for real-valued Fréchet differentiable functions on a normed space?

Using the Lagrange form for the remainder, Taylor's theorem can be stated as follows: Let $I\subseteq\mathbb R$ be an interval, $f\in C^{n+1}(I)$ for some $n\in\mathbb N_0$ and $s,t\in I$ ...
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66 views

Derivative of $x^x$ and the chain rule

Rewriting $x^x$ as $e^{x\ln{x}}$ we can then easily calculte the ${\frac{x}{dx}}$ derivative as ${x^x}(1 + \ln{x})$. We need to use chain rule in form $\frac{de^u}{du}\frac{du}{dx}$. The question is ...
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Proof that $f[x_0,x_1,…,x_n,\epsilon,\epsilon]=\frac{f^{n+2)}(\eta)}{(n+2)!}$

Up to now i have the following rule for divided differences: Assuming $x_0 \le x_1 \le...\le x_n$ then If $x_0 \lt x_n$ then ...
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57 views

When is this sine function differentiable at all points?

I have a hard time solving these kinds of problems, here is an example. For which values of a and b is the following function differentiable at all points? $$f(x)=\sin(|x^2+ax+b|)$$ Thanks in advance. ...
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derivative of error function

How can I calculate the derivatives $$\frac{\partial \mbox{erf}\left(\frac{\ln(t)-\mu}{\sqrt{2}\sigma}\right)}{\partial \mu}$$ and $$\frac{\partial ...
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How would I differentiate an integral with bounds?

Let $\space f \space$ be a differentiable function of $\space x \space$. Now I know that for the following integral: $$I=\int f(x) \space dx$$ Clearly: $${dI\over dx}=f(x)$$ Since integration is ...
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88 views

What is $\lim_{h \to 0} \frac{e^{x+h} - e^x}{h}$?

What is the $\displaystyle \lim_{h \to 0} \dfrac{e^{x+h} - e^x}{h}$? I'm not sure how to go about getting the solution.
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If $f(x) = -2\sin(x)$ then $f′(x)$ equals what?

If $f(x) = -2\sin(x)$ then $f′(x)$ equals what? A: $2\cos x$ If $f(x) = (15)^x$ then $f′(x)$ = ? A: $(15)^x \ln (15)^x$ Are my solutions correct?
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Why is continuity permissible at endpoints but not differentiability?

Differentiable at endpoints? cause of differentiation only on an open set. Admittedly, there are some questions and answers as to why a function defined on a closed interval is not differentiable on ...
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Prove $\log u > \frac{u - 1}{u}$ for $u > 1$

How to prove that for $u > 1$ $$\log u > \frac{u - 1}{u}$$ without using integrals? I think I'm supposed to use derivatives or Taylor's theorem, as the exercise comes from a lecture about these ...
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Showing existence of a partial derivatives

How would one show that that $$f(x,y) = \frac{xy(x^2-y^2)}{x^2+y^2}$$ for $(x,y) \neq (0,0)$ and $f(x,y)=(0,0)$ if $(x,y)=(0,0)$ has second order partials but $f_{xy}(0,0) \neq f_{yx}(0,0)$. I was ...
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Integration and differentiation of Fourier series

I am interested in the properties of Fourier series under integration and differentiation, and I've noticed a "strange" phenomenon. Suppose I have a Fourier series which I Integrate, and suppose that ...
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derivative of 2 dimensional integral

Let $f: \mathbb{R}^3 \rightarrow \mathbb{R}: (x,y,t) \mapsto f(x,y,t)$ a derivable function in every direction. Define $\mathfrak R_{\alpha}(u,t) := \int_{L(\alpha,u)} f(x,y,t) d(x,y)$ met ...
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17 views

Given a normed space $X$ and $A:X\to\mathbb R$, how can I compute the second Fréchet derivative of $f(t):=A(x_0+th)$ for some $x_0,h\in X$?

Let $(X,\left\|\;\cdot\;\right\|)$ be a Banach space and $A:X\to\mathbb R$ be Fréchet differentiable, i.e. $\exists{\rm D}A:X\to\mathfrak L(X,\mathbb R)$$^1$ with $$\lim_{\left\|h\right\|\to ...
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48 views

$f$ is a twice-differentiable function, prove there is some $x\in (-1, 1)$ such that $f '' (x) = 0$

Suppose $f: \mathbb R \to \mathbb R$ is a twice-differentiable function and that $f(-1) = -1,\; f(0) = 0$ and $f(1) = 1$. Prove that there exists some $x \in (-1, 1)$ such that $f''(x) = 0$.