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

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Does differentiation of $f(x)=\log(x)$ yield two different results?

The two different results are :$\frac{1}{x}$ and $\frac{-1}{x}$. I read in my book that: $$\frac{d(\log x)}{dx}=\frac{1}{x}$$ where $x>0$ And: $$\frac{d(\log(-x)}{dx}=\frac{1}{x}$$ where ...
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3answers
45 views

Total derivative notation help

consider the function $$f = f(x(t),y(t))$$ I know that its total derivative wrt t is $$\frac {df}{dt} = \frac {\partial f} {\partial x} \frac {dx}{dt} + \frac {\partial f}{\partial y} \frac ...
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1answer
44 views

Find the derivative of $y = x^{1/2}$ by using differentiation from first principle. [duplicate]

For this question, I tried to apply the derivative limit formula on it but I have a problem with the square root there: $$\lim_{\Delta x \rightarrow 0}\frac{\sqrt{x+\Delta x}-\sqrt x}{\Delta x}$$ If I ...
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7answers
375 views

How do I simplify and evaluate the limit of $(\sqrt x - 1)/(\sqrt[3] x - 1)$ as $x\to 1$?

Consider this limit: $$ \lim_{x \to 1} \frac{\sqrt x - 1}{ \sqrt[3] x - 1} $$ The answer is given to be 2 in the textbook. Our math professor skipped this question telling us it is not in our ...
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2answers
34 views

How to differentiate $ y=\sin^2(2x)\cos(x) $?

I was solving some A Level past papers and I came across this question. We have the equation of the line $ y=\sin^2(2x)\cos(x) $ for $ 0\leq x \leq \frac{\pi}{2} $ and there is a maximum point M. We ...
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1answer
16 views

Taking the derivative of a function of a convex combination of vectors, $f((1-t)x + t\cdot y)$

Let $f$ be a differentiable function, $x\not = y$ and vectors (say in $\mathbb{R}^n)$, and define $g:(0,1] \to \mathbb{R}$ by $$ g(t) = f((1-t)x + t\cdot y) $$ How would I differentiate this with ...
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2answers
27 views

Need clarifying on basic derivatives of natural log/e

So here's the question: Find the derivative: $ y= e^{\cos(x)}$ Hint: This is a combination of the chain rule and the natural log. The derivative is $(\ln a)(a^{f(x)}) * f'(x)$ So ...
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2answers
35 views

Find inflection points of the function $\arctan{\frac{x^2}{x^2-4}}$ [closed]

Find inflection points of the function $f(x) = \arctan{\frac{x^2}{x^2-4}}$ I found its second derivative and equated it to zero: $$f''(x)=\frac{4(3x^4-4x^2-8)}{(x^4-4x^2+8)^2}=0$$ How to find ...
2
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1answer
31 views

General clarification for derivative notation

I am a bit confused on the different notations of derivatives, could you help me clear it up? The following can be interpreted as: the total derivative of f wrt x, or equivalently, the derivative ...
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2answers
23 views

Sufficient conditions for applying Taylor theorem

Consider a real-valued function $f:\mathbb{R}\rightarrow \mathbb{R}$. Is assuming $f(.)$ twice differentiable at $a \in \mathbb{R}$ enough to apply the Taylor Theorem stating $$ ...
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2answers
49 views

Find if $\sqrt[4]{x^4+y^4}, \sqrt{x^4+y^4}$ are differentiable in $(0,0)$

Find if $$f(x,y)=\sqrt[4]{x^4+y^4}$$ $$g(x,y)=(f(x,y))^2$$ are differentiable in $(0,0)$. well, $g(x)$ is clearly $\sqrt{x^4+y^4}$, so I guess the answer will be similar to $f(x)$. ...
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1answer
19 views

Implications of bounded second derivative

Consider a real-valued function $f: \mathbb{R}\rightarrow \mathbb{R}$. Suppose we are said that the second derivative exists and is bounded in a neighbourhood of $x\in \mathbb{R}$. Does it imply that ...
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1answer
29 views

A partial derivative problem related with elasticity of substitution in Advanced Micro

Exe 3.8 Sorry, it is a problem that appears in Jehle and Reny Advanced Microeconomic Theory (3rd ed) exercise 3.8. But I think it's a partial derivative question. Letting $f_i(\mathbf{x})=\partial ...
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2answers
49 views

First derivative meaning in this case

If we have a function: $$f(x)=\frac{x}{2}+\arcsin{\frac{2x}{1+x^2}}$$ And it's first derivative is calculated as: ...
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4answers
39 views

Differentiabillity and continuity

If I have a function like $f(x)= \left\{\begin{array}{lr} 2, & \text{for } x>0\\ -2, & \text{for } x\leq0 \end{array}\right\}$ it is obviously not continuous in ...
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1answer
125 views

Prove that this function is differentiable

I came across this problem while I was studying for a preliminary exam and now I've devoted quite some time to it and can't figure it out. Any help would be greatly appreciated! Let $f : \mathbb R ...
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0answers
16 views

Explanation of Template Matching formula

Can someone please explain the formula f.) on OpenCV template matching Formula: Suppose template image is 3x4 and source image is 15x20 how would the mathematical operations follow...
3
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1answer
29 views

$f \in C^1$ defined on a compact set $K$ is Lipschitz?

Let $f: \Omega \subseteq \mathbb{R}^N \to \mathbb{R}^M$ be $C^1$, and $K \subseteq \Omega$. Prove that $f \mid_K$ is Lipschitz. Letting $x,y \in K$, I know that $f$ is loccaly Lipschitz, I ...
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2answers
50 views

Equivalence between definitions of derivative

Consider a function $F:\mathbb{R}\rightarrow \mathbb{R}$. I know this definition of derivative of $F$ at $x$: $$ \lim_{h \rightarrow 0} \frac{F(x+h)-F(x)}{h} $$ I found this definition of derivative ...
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1answer
45 views

Is $g$ is differentiable?

Let $f:\Bbb R\to \Bbb R$ be a differentiable function. Define $g:\Bbb R^2\to \Bbb R $ as $g(x,y) =f(\sqrt {x^2+y^2})$. Is $g$ differentiable? If we can show that one of the partial derivatives ...
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2answers
30 views

Differentiate without know formula $[\arcsin x]' = \frac{1}{\sqrt{1-x^2}}$

Is there any way how to get differentiate of $\arcsin x$ without memorize it? $$[\arcsin x]' = \frac{1}{\sqrt{1-x^2}}$$
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2answers
16 views

Finding the Equation of a Line Using Horizontal Tangents and Derivatives

I'm stuck on this one problem for my homework and it involves using given horizontal tangents to find an equation for a line using a generic polynomial. For the sake of following the rules of the ...
0
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1answer
74 views

Derivative of one function with respect to another

What is the derivative of $\tan^{-1}\left(\frac{\sqrt{1+x^2}-1}{x}\right)$ w.r.t. $\tan^{-1}\left(\frac{2x\sqrt{1-x^2}}{1-2x^2}\right)$ at $x=0$? Take ...
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0answers
8 views

Rewriting Black-Scholes differential equation

I am given the Black-Scholes PDV: $\frac{\delta V}{\delta t} +rS\frac{\delta V}{\delta S} +\frac{1}{2}\sigma^2S^2\frac{\delta^2V}{\delta S^2} -rV =0$ Now the following variable transformation takes ...
1
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2answers
53 views

Find function f(x)

Find function f(x), where: $$f(3)=3$$ $$f'(3)=3$$ $$f'(4)=4$$ $$f''(3) = \nexists$$ How to find function like this in general? What steps should I do?
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3answers
52 views

What type of continuous function will be differentiable?

I know that every differentiable function is continuous but converse is not true. So how I can say that a continuous function will be differentiable. That a continuous function will be differentiable ...
0
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2answers
20 views

A Limits Problem [closed]

Using the fact that $$\lim_{h \to 0}\dfrac{\sin(h)}{h} = 1$$ and $$\lim_{h\to 0}\dfrac{\cos(h)-1}{h}=0\text{,}$$ Compute the following limits: $\lim_{h\to 0} \dfrac{\sin(x+h)-\sin(x)}{h}$ ...
0
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1answer
43 views

if $|f'(x)|\le 4/5$ then is there a unique x such that $f(x)=x$

Let $f:\mathbb{R} \to \mathbb{R}$ be continuously differentiable and such that $|f'(x)|\le \frac{4}{5}$ for all $x \in \mathbb{R}$ then does a unique $x\in \mathbb{R}$ exists such that $f(x)=x$? My ...
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1answer
40 views

if $|f(x)-f(y)|\le |x-y|^{\sqrt 2}$ then is $f$ a constant function?

if $f: \mathbb{R}\to \mathbb{R}$ satisfies $$|f(x)-f(y)|\le |x-y|^{\sqrt{2}}$$ for all $x,y\in \mathbb{R}$ ,then is f increasing ,decreasing or constant? in my view ,it is clear that $|f(x)-f(y)|$ is ...
1
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0answers
108 views

If $f_1(k)$ and $f_2(k)$ reach their max. at $k_{m1}$ and $k_{m2}$, resp., show that $k_{m1} > k_{m2}$ in the following case

Let $k$ represent an integer value. We define function: $f_1(k)=k(1-k t)\frac{\Gamma(L-k,k a_1)}{\Gamma(L-k)}$, for $1\le k\le L-1$, with $kt \le 1$, and $a_1$, $t (<1)$ are some positive ...
0
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1answer
23 views

Does the function have derivatives at $x=0$

derivatives Hey, in this question I succeeded the first part. Part B: I fail to show that either have no derivative at the point $x = 0$. I try on the definition of derivative and lodged. Part C: ...
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2answers
21 views

Prove that a funtion is differentiable at zero

Prove that for a function $f:(-1,1)\rightarrow \mathbb{R}$ and for it holds that $\mid f(x) \mid\le x^2$. Prove that $f$ is differentiable in $0$ and that $f'(0)=0$. From the domen and $\mid f(x) ...
1
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1answer
81 views

Why do derivatives of functions exist?

Consider following function: $f(x)=x^2 \sin{\frac{1}{x} }$ if $x\neq 0$ and $f(0)=0$. Why does the derivative of $f(x)$ exist? Find the deriviative and determine whether or not it is continous. ...
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3answers
63 views

Geometric Interpretation of a “Near”-MVT

Going through Larson's Problem Solving Through Problems, I am asked to give a geometric interpretation of the result below. I have been sketching it, and only got so far as to note that there must be ...
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0answers
16 views

Partial derivatives of a function with conditions dependent on parameters

Not sure if that question title makes any sense, but here's my problem. I have a function $$ f(x,\alpha,\beta) = \begin{cases} {\frac{x-\alpha}{\beta-\alpha}} & {\alpha \leq x \leq \beta}\\ {0} ...
3
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1answer
119 views

Degree two homogeneous differentiable function is a quadratic form.

Let $f: \Bbb R^n \to \Bbb R^k$ be a ${\cal C}^2$ function such that $f(tx) = t^2f(x)$ for all $t \in \Bbb R$ and all $x \in \Bbb R^n$. Then there is a bilinear map $B: \Bbb R^n \times \Bbb R^n\to ...
4
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1answer
68 views

meaning of the notation f'(-x)

What does $f'(-x)$ essentially mean? $\frac{df(-x)}{dx}$, or $\frac{df(x)}{d(-x)}$, or $\frac{df(x)}{dx}|_{x=-x}$ ? I am not sure if all the options are different, though! :) EDIT 1: Let me ...
0
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1answer
42 views

Vector space of a sequence of scalars

Struggling slightly with the following question, not sure how to proceed: Let $c_{0}$ be the vector space of sequences of scalars $(a_{n})_{n\in\Bbb{N}}$ such that $a_{n} \to 0 $ as $ n \to \infty $. ...
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0answers
28 views

$f$ is differentiable on $[a,b]$, show that $f'([a,b])$ is an interval

Function $f$ is differentiable on $[a,b], a< b, a,b \in \mathbb R$. I need to show that $f'([a,b])$ is an interval. The standard approach would be to take two points from the interval (I assume ...
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0answers
19 views

Compute a derivative $\frac{df (a, c)}{dg(a, b)}$

I tried to compute the derivative $\dfrac{df(a, b)}{dg (a, c)}$ and wanted to check if what I did was legal. $\dfrac{df(a, b)}{dg (a, c)} = \dfrac{df}{da}*\dfrac{da}{dg} + ...
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1answer
33 views

Directional derivative of a piecewise defined function

Given $f(x,y)=\left\{\begin{matrix} \frac {x^2y}{x^4+y^2} & (x,y)\neq(0,0)\\ 0& (x,y)=(0,0) \end{matrix}\right.$ I need to calculate the directional derivative at the point (0,0) in the ...
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0answers
28 views

partial deravtive of $\sin xyz - { 1 \over z-xy} = 1$

Given z(x,y) and $\sin xyz - { 1 \over z-xy} = 1$. How to calculate $z_x(0,1)$ ? Let $$F=\sin xyz - { 1 \over z-xy} - 1 = 0 $$ $$z_x= - \frac{F_x}{F_z}= -{y(z+xz_x) \cos xyz + \frac{z_x - y}{(z-xy)^2} ...
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0answers
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Derivation of the hypergeometric function $\frac{\partial {}_{3}F_{2}(a_{1}, a_{2}, a_{3}; b_{1}, b_{2}; \frac{1}{z})}{\partial z}$

We know that the first order derivative of the generalized hypergeometric function ${}_{3}F_{2}(a_{1}, a_{2}, a_{3}; b_{1}, b_{2}; z)$ is expressed as follows: \begin{equation} \frac{\partial ...
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0answers
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Computing derivative of a composition.

I have to compute the derivatve $f(g(u,v))$, where $$f(x,y)=x^2+y^3+3xy^2+5,\; \; g(u,v)=(u-v,u^2+v^2)$$ in the point $(u',v')=(1,1)$. Can anyone explain to me how do I find such derivatives, a ...
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2answers
320 views

Derivative by definition

I'm trying to find the derivative by definition of the following function: $f(x)=\sqrt{|x|}\sin(x)$ I know that by definition: $$ f'(x)=\lim_{h\to0}\frac{\sqrt{|x+h|}\sin(x+h)-\sqrt{|x|}\sin(x)}{h} ...
0
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1answer
55 views

Why does this partial derivative of a summation work?

I'm trying to take the partial derivative of $-\sum\limits_{i=1}^n \frac{(x_i-\mu)^2}{2\sigma^2}$ with respect to $\mu$. The correct answer is $\sum\limits_{i=1}^n \frac{x_i-\mu}{\sigma^2}$. It ...
1
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2answers
33 views

Verifying differential equations (without substitution or integration)

I am aware that several similar threads exist on this forum, however, my particular query is different from any previous question I've seen here. All other answers have a 'y'term on both sides, and ...
2
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3answers
45 views

Textual explanation of a derivative

In the book Structure and Interpretation of Computer Programs, there is an interesting example on how one might explore symbolic data in programming. They used the differentiation algorithm. That is, ...
0
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0answers
19 views

Minimising logistic loss function to find optimal matrix

Please take a look at this paper on classifying triples (re link prediction): http://arxiv.org/pdf/1510.04935v2.pdf The question is about how to solve equation 2 using stochastic gradient descent. It ...
0
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1answer
30 views

Find the minima of $I(a, b) = \int_0^1 (ax + b-x^2)^2 dx$

$I(a, b) = \int_0^1 (ax + b-x^2)^2 dx$ How to find the minima of $I(a,b)$? My idea is like this, $ I(a,b) = \frac13(ax + b -x^2)^3 \cdot (\frac12 ax^2 + bx - \frac13 x^3) |_0^1$ This is the first ...