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

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1answer
43 views

Is my calculation right for differentiability?(with complete resolution if right)

In the following completed example I ask if it is done right. $$f(x,y)=\begin{cases} \frac {2x^2y}{x^2+y^2} \mbox{for} (x,y)\neq (0,0) \\0 \mbox{for} (x,y)=(0,0) \end{cases} $$ Now and the partial ...
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2answers
113 views

Not sure how to differentiate implicitly using parametric equations…

I am not sure (not taught before explicitly) how to apply implicit differentiation on parametric equations when I am solving the question posted below. Question Two positive numbers $x$ and $y$ ...
3
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2answers
148 views

Partial Derivative v/s Total Derivative

I am bit confused regarding the geometrical/logical meaning of partial and total derivative. I have given my confusion with examples as follows Question Suppose we have a function $f(x,y)$ , then ...
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0answers
39 views

Does linearity imply differentiability?

Is a linear function differentiable at every point by definition?
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3answers
52 views

limitations problem needing help to solve

$$\lim_{k\to x}\frac{ \sin^2 k - \sin^2 x}{k - x}$$ I have tried to solve it over and over but couldnt. I will be very happy if someone can show me the way to solve this .
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1answer
102 views

What is the (rigorous) reason that the derivative of $|x|$ does not exist at $x=0$?

Let $g=|x|$. Then, the derivative at $c=0$ is given by: $$ g'(0) = \lim_{x \to 0} \frac{|x|}{x} $$ which is either $+1$ if $x$ comes from the positive $x$-axis or $-1$ if $x$ comes from the negative ...
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6answers
72 views

How do you find the derivative of $2^{\sin(\pi x)}$?

I don't understand how to take the derivative of this expression. $$y=2^{\sin (\pi x)}$$
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2answers
183 views

Differentiation with respect to variable

After scouting around I've had no luck in finding answers to my question which is relatively simple for alot of you. Before jumping straight to the question I 'd like to clarify that I'm not only ...
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3answers
205 views

Is there a function whose derivative is $|x|$?

Is there a function $y=f(x)$ such that $$\frac{df}{dx}|_{x=a} =|a|$$ for all $a\in \mathbb R$? I'm in a debate with my friend over it and we are stuck
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0answers
45 views

Derivative : tangent line and multiplication of derivative

Could anyone give me a hint how to prove the following statements ? I suppose that I have some general ideas of pictures of functions satisfying the condition should look like. But it seems impossible ...
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0answers
23 views

Error of Riemann sum is $a/n + o(1/n)$ [duplicate]

A problem from an old qual: For $f$ of class $C^2$, find $a$ such that $$\int_0^1 f(t)dt-\frac1n\sum_{k=1}^{n-1}f\left( \frac {k}{n}\right)=\frac{a}{n}+o\left(\frac1n \right).$$ If we divide ...
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1answer
32 views

how the ratio of two functions change…what am I doing wrong?

For $s \in \{1,\dots,T-2\}$, let $g(s) := \frac{f(s+1)}{f(s+2)} = \frac{\sum_{t=s+1}^{T} \frac{0.99^{t-1}}{1 + \text{exp}\left(\frac{t-1}{3} - 9\right)} \frac{1}{t}}{\sum_{t=s+2}^{T} ...
2
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3answers
40 views

Differentiating a function by simplification.

If we consider a function: $f\left(x\right)=\dfrac{x-1}{2x^2-7x+5}$ This function is not defined at x=1 and x=5/2. So if we differentiate this function by u/v method we have: ...
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1answer
39 views

Proving the correctness of alternative derivative definition

We know that the derivative exists as $f'(x)=\lim_{h \to 0}\dfrac{f(x+h)-f(x)}{h}$. Using this, how can we prove that it is $f'(x)=\lim_{h \to 0}\dfrac{f(x)-f(x-h)}{h}$ as well? I tried the definition ...
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2answers
69 views

How to prove that a derivative of a formula equals to another formula.

If $u= \ln(\tan x+\tan y+\tan z)$ prove $$\sin 2x \dfrac{du}{dx} + \sin 2y \dfrac{du}{dy} + \sin 2z \dfrac{du}{dz}=2 $$ My answwer was like this: $$u' =\dfrac{ 1}{\tan x+\tan y+\tan z} \cdot( ...
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1answer
70 views

How to find the third derivative of the function given by $\rho=\tan(\rho+\phi)$

Given that $\rho=\tan(\rho+\phi)$, how to find $\frac{d^3 \rho}{d\phi^3}$? Answer is: $$-\frac{2(5+8\rho^2+3\rho^4)}{\rho^8}$$ I start from $$\rho=\tan(\rho+\phi) \tag{1} $$ and use ...
2
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0answers
257 views

Differentiating $\arcsin(x)$ using first principle method.

Q. Differentiate $\sin^{-1}(x)$ using first principle (delta) method. I did this the following way: $$y=\sin^{-1}(x)$$ $$\therefore\frac{dy}{dx}=\lim_{h\to ...
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0answers
24 views

Limits and Displacement Relationship

So my Calculus 3 professor began our class by introducing vectors and such, instead of delving into 3D integrals and such, which is understandable, he just wants us to develop a understanding of 3D ...
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2answers
164 views

Question about the limit definitions of derivative and definite integral

Actually this question may look simple and basic, but it is about something which bugs me for a long time, since when I took my first calculus classes. The limit $\lim_{x \to a}f(x) = L$ is defined ...
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1answer
32 views

Is it Preference relation?

I need to check if the relation $ \succeq ( \space \succeq \space \subset X × X, \space X=VB[0,1] ) $ define as below $$ f \succeq g \Longleftrightarrow Var(f+g) \geq Varf $$ is preference ...
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2answers
22 views

Finding the derivative with the limit definition

$f(x) = \frac{2}{\sqrt{x}}$ at $x = 9$. I plugged 9 in for all the places x showed up for in the limits definition and I end up with $$\frac{2(3 -\sqrt{9+h})}{3h \sqrt{9+h}}$$ I'm unsure of what to do ...
3
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1answer
454 views

Finding the derivative of a rational function with limit definition

The problem is to find the derivative of $f(x) = \frac{3x}{x^2+1}$ at $x = -4$ using the limit definition, $$ f'(x) = \lim_{h\to 0}\frac{f(x+h)-f(x)}{h} $$ Progress I plug in $-4$ for $x$ when using ...
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1answer
38 views

Confusion with derivative rule and solving with limit definition

$$f(x) = \frac{4x + 1 }{x -2}\text{ at }x = 5.$$ When I try to find the derivative by the definition of a limit I end up with $-1$. I tried doing it with the rule and I'm sure I went wrong somewhere ...
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1answer
36 views

Series representation of function with fractions, logarithms, squares and cosines.

I'm looking for a series representation for $$\dfrac x{x^2+(\log \cos x)^2}$$ Where $x\in(0,\pi/2)$ Note: Both finite and infinite series are accepted. I have tried taylor series, but it requires ...
3
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1answer
69 views

$n$th erivative of $f(x)^{n+1}$

Hello, I'm trying to find the $n$th derivative of a function, where $n\in\mathbb N$ $$\frac{d^{n-1}}{dx^{n-1}} \!\!\left[f(x)^n\right]$$ I'm looking for some kind of sum or product (or nesting ...
2
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1answer
60 views

Strange claim by WA involving nth derivative

Playing a bit around with WA i found this Namely: $$\frac {d^n}{d^nx} \left(\frac x{f(x)}\right)^{n+1}=x\left(\frac 1{f(x)}\right)^{n+1}(2)_n$$ For $n\in\mathbb{N_0}$ and $n+1\ne x$ and $x \ne 0$ ...
2
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1answer
51 views

Derivative of an Inverse Function

Can someone please give me a simple proof of this- If $f$ is differentiable on an interval containing $c$ and $f'(c) \neq 0$, then $f^{-1}$ (inverse of $f$) is differentiable at $f(c)$. I can see ...
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0answers
37 views

help with this derivative involving transformation matrix

I am working on an image analysis problem where I have a cost function which has the following term: $$ U = f(M(p)) $$ Here $f$ is an image which is sampled at discrete locations $p$. $M$ is a rigid ...
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3answers
38 views

When to use implicit differentiation?

When a problem is expressed like this $F(x,y)$ then you are asked to find $F_x(x,y)$ My understand is that means you need to find the partial derivative of $x$, which means $y$ is held and treated as ...
2
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2answers
45 views

Question on Partial Derviatives

For function $f(x,y) = x^2 y$ The partial derivatives for $x$ is $2.x.y $. I'm new to such math equation and i'm learning them now. May i know why is it so? Thanks!
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2answers
55 views

Both partial derivative

Let $f:\mathbb{R}^{2} \rightarrow \mathbb{R}$, $$f(x,y) = \begin{cases} \dfrac{xy^2}{x^2+y^4}, & x^2+y^2 \neq 0, \\ 0, & x=y=0. \end{cases}$$ Show that both partial derivatives exist at ...
0
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1answer
46 views

Partial Derivative of the one variable function

This is from my exam: 1) Calculate partial derivative $f'(10)$ of the function: $$f(x)=\frac{1-\log x}{1+\log x}.$$ This is a function of only one variable, why do they use the term 'partial' ? Are ...
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3answers
162 views

Find constant $k$ so the line $x+y=k$ is normal to the curve $y=x^2$ [closed]

For what value of the constant $k$ is the line $x+y=k$ normal to the curve $y=x^2\;?$ Can anyone help me understand how to approach this problem? Thanks in advance.
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3answers
154 views

Proving a second derivative

Given that $$y = \sin^3 x + \cos^3 x$$ prove that $$\frac{d^2 y}{dx^2} = \frac{3}{2} (\cos x + \sin x)(3 \sin 2x - 2)$$ I began with differentiating the equation as it is and it took me around ...
3
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2answers
47 views

Second derivative expression

I have $f:\mathbb R^n\to \mathbb R$ and $\gamma:\mathbb R \to \mathbb R^n$, which are both $\mathrm C^2$. Considering $g=f\circ \gamma$, how could I express $g''$, second derivative of $g$ in terms of ...
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2answers
73 views

Using the mean value theorem to prove inequalities

Using the Mean Value Theorem, show that for any $t>0$, $$\left|e^{-x^2/2t}-e^{-y ^2/2t}\right|\leqslant \frac{|x-y|}{t}$$ for all $x,y$ with $|x|,|y|\leqslant 1$. My attempt. Without loss of ...
2
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0answers
48 views

How good an approximation to the derivative is an arc-length based approximation?

Note - my original definition below was wrong. I hope this replacement is better. The usual approximation to $f'(x)$ with step size $h$ is $D_h(f, x) = \frac{f(x+h)-f(x)}{h} $. This has so many nice ...
0
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1answer
51 views

If $|f(x)|<x^2$ for $|x|<1$, then show that $f'(x) $ exists and find it [closed]

I can intuitively guess the answer is 0 because if we graph of f(x) must be below x^2 looking for a clean and more logical attempt
2
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0answers
49 views

Taking partial derivatives over multiple summations

I have the following equation obtained from one of the models. $\mathcal{H} = \sum\limits_{D} \sum\limits_{W}n(d,w)\sum\limits_{Z} p(z|d,w)[\log{p(d)}+\log{p(z|d)}+\log{p(w|z)]}$ I need to take ...
2
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0answers
42 views

Determining $f^{-1}(3)$ without knowing $f^{-1}(x)$ but given $f(1)=3$ and $f'(x)>0$.

I have a continuous function $f(x)$ and I want to find $f^{-1}(3)$, but I can't find $f^{-1}$ directly. I know that $f(1)=3$ and $f'(x)>0$ for all x. Because the function is continuous and always ...
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4answers
41 views

Find $\frac{dy}{dx}$ for $x=2\theta+sin2\theta$ and $y=1-cos2\theta$

The parametric equations of a curve are $$x=2\theta+\sin2\theta,\:y=1-\cos2\theta.$$ Show that $\frac{dy}{dx}=\tan\theta$. I can use the chain rule to get ...
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2answers
144 views

Integrable combinations - I can't seem to arrive at the given answer

I need help! I can't seem to arrive at the answer given in our textbook. I'm new here, so I really need help. The instruction says that I need to solve this D.E by recognizing integrable ...
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0answers
37 views

how to find the optimal function with lagged cost? (calculus of variations)

I need to find the function $b( )$ that maximizes this guy ($c()$ and $\beta()$ are functions too, and $c()$ is convex): $$\int_{0}^{T} \! e^{-\delta v}\beta(v) \left[\int_{0}^{v} b(s) \; ds - ...
2
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2answers
110 views

Trying to understand “derivative or Jacobian of smooth map”

From some lecture notes I am trying to puzzle through .... "... the derivative or Jacobian of a smooth map $f: \mathbb{R}^m \rightarrow \mathbb{R}^n$ at a point $x$ is a linear map $Df: \mathbb{R}^m ...
3
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1answer
96 views

Calc II - Definite integral of sqrt(t^2 + t) from 2x to 1?

How do I find $$\int_1^{2x}\sqrt{t^2 + t}$$ with only knowledge from a Calculus I course? I've tried plugging this puppy into Wolfram Alpha and other integral solvers, which report it as solvable ...
2
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3answers
279 views

Limit of a Riemann Sum and Integral

I've been trying to solve this problem, but I haven't been able to calculate the exact limit, I've just been able to find some boundaries. I hope you guys can help me with it. Let $f:[0,1] \to ...
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1answer
36 views

Discretization of v*(du/dx)

I am trying to discretize the term: $$\underline{v}\frac{d\underline{u}}{dx}$$ using finite differences or evaluate $$\int_{\Gamma}\underline{v}\frac{d\underline{u}}{dx}.\underline{n}d\Gamma$$ ...
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1answer
79 views

Does the symmetric decreasing rearrangement of a smooth function preserve smoothness?

Let $A\subset \mathbb{R}^n$ a Borel set of finite Lebesgue measure. They define $A^*$ to be the ball centered at 0 with the same measure that $A$. The symmetric-decreasing rearrangement of ...
2
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1answer
46 views

If $f$ has derivative at $1$ and $\lim_{h \to 0} {\frac{f(1+h)}{h} }=1$, then $f'(1)=0=f(1)$

I need to prove that if $f(x)$ has derivative at $x=1$ and if $\lim_{h \to 0} {\frac{f(1+h)}{h} }=1$. then I need to prove that $f'(1)=0$ $f(1)=0$. It's pretty obovious if using arithmetic of limits, ...
0
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1answer
84 views

Proving that the exponential function is its own derivative, using the limit definition of $e$

I saw the proofs on the derivative of $\frac{d e^x}{dx}=e^x$ from here and the one that was intriguing was this : $$e^x:=\lim_{n\to\infty}\left(1+\frac{x}{n}\right)^n \implies \frac{d(e^x)}{dx} = ...