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

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4answers
54 views

Integration by parts - hint

I'm stuck on a passage on my textbook: $$ \int \frac{1}{(1+t^2)^3} dt = \frac{t}{4(t^2+1)^2}+\frac{3}{4} \int \frac{1}{(t^2+1)^2} dt$$ I know that it should be easy but I just can't figure out what ...
2
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2answers
39 views

Find the $\Delta y$ of $f(x)={1 \over x^2}$; $x=2; \Delta x = 0.01$

Find the $\Delta y$ of $f(x)={1 \over x^2}$; $x=2; \Delta x = 0.01$ when $\Delta y = f(x+ \Delta x) - f(x)$ So here's what I did: $$\Delta y = f(x+ \Delta x) - f(x) \\ \Delta y = {1 \over (x+ ...
0
votes
1answer
21 views

Given that $f_1$ and $f_2$ are differentiable, find $Df(x_o)$ in terms of $f_1'(x_o)$ and $f_2'(x_0)$. [closed]

this is my first time asking a question on here but I am completely stuck. to answer this question, we need to use linear approximation and I'm just confused Let $f: \mathbb{R} \to \mathbb{R}^2$, ...
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0answers
13 views

Behavior of $J/I$ w.r.t $m_1$, $I=\int_{m_1k}^{\infty} t^{N-k}e^{-t} dt$ and $J=(m_1k)^{N-k} e^{-m_1k}-\int_{m_1k}^{\infty} \log(t) t^{N-k}e^{-t} dt$

Let us define $I=\int_{m_1k}^{\infty} t^{N-k}e^{-t} dt$ and $J=(m_1k)^{N-k} e^{-m_1k}-\int_{m_1k}^{\infty} \log(t) t^{N-k}e^{-t} dt$. We assume that $m_1 \ge 0$, $k \ge 0$ and $k \le N$. Using the ...
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3answers
72 views

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 ...
1
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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 ...
0
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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 ...
7
votes
7answers
377 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
35 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 ...
0
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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 ...
0
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2answers
36 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
votes
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 ...
1
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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 $$ ...
1
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2answers
52 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 $f(x)$. ...
0
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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 ...
0
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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 ...
4
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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 ...
1
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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 ...
0
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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 ...
1
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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 ...
-1
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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
53 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
44 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 ...
1
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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: ...
-1
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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 ...
1
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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
votes
1answer
120 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
votes
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
43 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 $. ...
0
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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 ...
0
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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} + ...
0
votes
1answer
35 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 ...
0
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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} ...
-1
votes
0answers
19 views

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 ...
0
votes
0answers
19 views

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 ...
1
vote
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
votes
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 ...