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

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0
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1answer
40 views

Locate and classify stationary points

Locate and classify as maxima, minima or saddle point the stationary points of the surface given by the equation $$z=(5x+7y-25)e^{-(x^2+xy+y^2)}.$$ Stationary points are the points where the gradient ...
1
vote
1answer
72 views

Jacobian of mapping

Let's say we're in $\mathbb{R}^n \times \mathbb{R}^n$ and we have the identity mapping $f:\mathbb{R}^n \times \mathbb{R}^n \rightarrow \mathbb{R}^n \times \mathbb{R}^n$, $f(x,y) = (x,y)$. What I want ...
3
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1answer
30 views

Question about differential operators

Say $N = ab$. How can I express $\frac{d}{dN}$ in terms of $\frac{d}{da}$ and $\frac{d}{db}$?
1
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1answer
22 views

Graphing derivatives of non-function equations

Is it possible to graph the derivatives of equations that fail the vertical line test? Such as a circle, a folium of descartes, an asteroid, etc?
0
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1answer
28 views

Special Derivative Function

I have a function $f(x)$ which I would like to have the derivative with respect to $x$. How can I get the derivative of the following function with respect to $x$? $$f(x) = \log(1-z^{e^{y^{T}x}})$$
0
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4answers
126 views

Find the absolute maximum/minimum values of S(t) where S'(t) is a quartic function with lots of horrible decimal places.

So I have a problem where I'm to find the absolute maximum and minimum values of the following function... $S(t) = -0.00003237t^5 + 0.0009037t^4 - 0.008956t^3 + 0.03629t^2 -0.04458t + 0.4074$ ...
9
votes
4answers
1k views

Is the determinant differentiable?

I was wondering, given an $n\times n$ square matrix with $n^2$ many entries, the function $\det:\left(a_1,a_2,\ldots,a_{n^2}\right)\to \textbf{R}$ which gives the determinant where $a_{k}$'s are the ...
1
vote
1answer
64 views

Find speed of an aircraft flying towards an observer

An aircraft is flying towards an observer at an altitude of $2000$ m. When the angle of observation, $z$, (between the ground and the aircraft) is 30 degrees, the rate of change of this angle is $2$ ...
6
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3answers
96 views

Limit of $x \ln^2|x|$ when $x\to 0$

I want to evaluate this limit :$$\lim_{x\to 0}x\ln^2|x|$$ I wanted to use L'Hôpital's rule for this: $\lim\limits_{x\to0-}\frac{\ln^2|x|}{\tfrac{1}{x}}$, but I don't know how to differentiate the ...
1
vote
1answer
43 views

find the maximum of the function F under the condition $ \sum_{i=1}^N x_i = 1$

Let F a function of $ \mathbb{R} ^N_+ \rightarrow \mathbb{R}$ defined as : $$F(x_1,..,x_N)= - \sum_{i=1}^N x_i log(x_i) , x_i \gt 0$$ How can i find the maximum of the function F under the ...
1
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1answer
29 views

Related rates of change - concentric spheres

Two concentric spheres each have an initial volume of 0. Their radii are increasing at 3mm/s and 5mm/s respectively. Calculate the rate at which the volume between the spheres is changing after 4 ...
1
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1answer
54 views

The derivative of $x^TAx$ w.r.t $t$

Suppose $P = x^TAx$ How to find $\frac{dP}{dt}$? if $x' = Bx$ , where $B$ has the same dimension as $A$. How to find the final answer? my answer is: $$\frac{dP}{dt} = 2[(A+A^T)x]x' = ...
0
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1answer
57 views

Showing the Clairaut theorem in higher dimensions — partials commute

Suppose $f$ has all partial derivatives up to and including $k$ and all of these partials are continuous. Prove that if $\sigma$ is a permutation on $n$ letters (any reordering), then ...
0
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2answers
39 views

Show that $f$ is everywhere differentiable and the partials commute

Take the function $$ f(x,y) = \begin{cases}\frac{x^3y -xy^3}{x^2+y^2} & (x,y) \neq (0,0) \\ 0 & (x,y) = (0,0) \end{cases}. $$ Show that it is everywhere differentiable and that $D_{1,2}f(0,0)$ ...
2
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1answer
411 views

Finding Tangent line for a Graph with the Natural Log

I'm really confused on how my professor did this problem. Any in depth explanation would be awesome. Thanks for your time.
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2answers
39 views

$f(x)=-4x^2+11126x-62516$. Time and how many

The question is this I've been trying to get my head around this but simply cannot and am hoping you might get me going. Q: The Store is open from $8$ am-$8$ pm every single day. $X$ represents the ...
1
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2answers
78 views

How to find the second derivative?

I use this article from Wikipedia to build it in my program. How to find the second derivative in $(x_i, y_i)$ point of this cubic interpolation, if I know other $(x_j, y_j)$ points?
4
votes
1answer
84 views

How does the chain rule work for more than one variable?

I know that that $$\dfrac{d\sqrt{x}}{dt} = \dfrac{d\sqrt{x}}{dx} \dfrac{dx}{dt}$$ In this equation there you only have 1 variable, namely $x$. But why is the following correct?: $$T = \frac{1}{2} ...
0
votes
1answer
53 views

Partial derivatives of $f(x,y)=\sqrt{|xy|}$

$f(x,y)=\sqrt{|xy|}$ First question: How to find $f_x(0,0)$ and $f_y(0,0)$? I have figured out this using definition - Both are $0$. My next question is: How to show that $f_x(0,0)$ and $f_y(0,0)$ ...
2
votes
2answers
201 views

If $f$ is twice differentiable then $f^{-1}$ is twice differentiable

$f:(a,b) \rightarrow (c,d)$ is a bijection and $f$ is differentibale with $f'(x) \neq 0$ for all $x \in (a,b)$, then $f^{-1}$ is also everywhere differentiable. Show that if $f$ is twice ...
1
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2answers
62 views

Are these derivatives correct??

Take take the function defined as $$f(x) = \left\{ \begin{array}{ll} exp(\dfrac{-1}{x^{2}}) & \mbox{if } x \neq 0 \\ 0 & \mbox{if } x = 0 \end{array} \right. $$ Now I am asked to check ...
1
vote
1answer
26 views

Identity concerning Bessel functions

I would like help showing the following is true: $$\frac{d}{dx}[x^{-\alpha}J_{\alpha}(x)] = -x^{-\alpha}J_{\alpha+1}(x).$$ I can show $\frac{d}{dx}[x^{\alpha}J_{\alpha}(x)] = ...
3
votes
0answers
42 views

Integration over time by having derivation

Assume we want to find the following integration: \begin{equation}\int_{t=0}^{\infty} p(t)dt\end{equation} where $p(0)=p$ and also $$\frac{dp(t)}{dt}=-p(t)(1-p(t))\mu$$. Is there any easy way to ...
1
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1answer
221 views

Can Moore–Penrose pseudoinverse solve for underdetermined linear system?

Thanks for reading my thread. I am thinking, many of us know that Moore–Penrose pseudoinverse can solve for overdetermined system $Ax=b$, where $x=(A^TA)^{-1}A^Tb$; for exmplae the linear regression ...
1
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1answer
62 views

Showing a function has first and second partials everywhere and continuity of the partials

Let $f \left( \begin{array}{ccc} x \\ y \end{array} \right)= \begin{cases} xy\dfrac{x^2-y^2}{x^2+y^2} & \mbox{if $(x,y) \neq (0,0)$}\\ 0 & \mbox{if $(x,y) = (0,0)$} \end{cases}. $ I ...
1
vote
4answers
52 views

Show limit of $n\log(1+ \frac{x}{n})$ exists

How would I do this question? The fact that $y \rightarrow \log(1+y)$ tells me that: $$\lim_{h \to 0} \frac{\log(1+0+h)-\log(1)}{h}$$ tends to a existing limit. How do I use this for my answer??
2
votes
5answers
72 views

The Notation for Derivatives

"The derivative of a sum is the sum of derivatives" Above theorem can be mathematically expressed as: $$h'(x)=f'(x)+g'(x)$$ where $f(x)$ and $g(x)$ are two differentiable functions. What is the ...
5
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1answer
89 views

Showing that $|f^{(n)}| \le n!n^n$ and then making this result sharper

Ahlfors: Show that the successive derivatives of an analytic function at a point can never satisfy $|f^{(n)}(z)| > n!n^n$. Formulate a sharper theorem of the same kind. Attempt for Part ...
7
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0answers
126 views

Higher Order Derivative Proof .

I would appreciate if someone could check over my proof for this question and advise me if it is correct. My attempt so far; Now as $f$ is k times differentiable , it taylor series about $x_{0}$ ...
2
votes
0answers
65 views

Interchange of limiting operations (question from an engineer)

I need to clarify when are the below operations valid. If possible, please link me to the related theorems, where I can find details. 1- Given a double integral \begin{equation} \int_{X}\int_Y ...
0
votes
3answers
53 views

Proving Differentiability rigorously

Assume that real function f is differentiable at $x_0$ with $f'(x_0)$ >0. How would one show that there exists a $\delta$>0 such that $$ f(x)>f(x_0) $$ for all x in between $x_0$ and $ x_0 + ...
4
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1answer
80 views

Differentiability implies continuity — possibly pedantic question about the common proof

The common proof that differentiability implies continuity arrives at this limit: $$\lim_{x\to a} [f(x) - f(a)] = 0$$ I'm failing to see the simple justification for moving to the next step, which ...
1
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1answer
60 views

One-sided Derivative Question

Let's say we define $$D_{+}f(x):=\lim_{h\to 0^+}\frac{f(x+2h)-f(x+h)}{h}$$ to be the "right-handed" derivative. This way the function does not have to exist (or equal what it 'should') at the point ...
1
vote
1answer
58 views

Let P(x) be a polynomial of degree 4 , having extremum at $x=1,x=2$ and $\lim_{x\to 0}\frac{x^2+P(x)}{x^2}=2$ Then the value of P(2)

Let P(x) be a polynomial of degree 4 , having extremum at $x=1,x=2$ and $\lim_{x\to 0}\frac{x^2+P(x)}{x^2}=2$ Then the value of P(2) is _______ I worked out the limit using L'Hospital got a relation ...
2
votes
3answers
256 views

What is defined by rate of change at a single point?

Rate of change measures how fast a process is going when it moves from one point to another. It measures the change of, say, $Y$ when $X$ moves from $X$ to $X + \Delta X$. But my problem arises when ...
0
votes
3answers
107 views

Where did the linear approximation/linearization formula come from?

Where did the linear approximation/linearization formula come from? I understand that it takes root in the point-slope form and slope intercept form of a linear equation, but I don't understand where ...
0
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0answers
29 views

Intermediate value of the derivative.

Hi all what would the best way be to approach this question? I tried using the hint but I can't seem to formulate an answer for the fist part. Any help for the first and second parts of the question ...
1
vote
1answer
41 views

Differentiation of $u^{T}Su$

I want to differentiate $u^{T}Su$ wrt $u$ where $u$ is $n$ x $1$ and $S$ is $n$ x $n$matrix . So I did the following . Since $u^{T}Su$ is a number , I wrote its expression ie $$ f = ...
0
votes
0answers
12 views

Derivatives of functions defined implicitly

Let $f$ and $g$ be functions of one real variable and define $F(x,y)=f[x+g(y)]$. Find the formulas for the partial derivatives of $F$ of first and second order, expressed in terms of derivatives of ...
30
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4answers
2k views

Is it necessary that every function is a derivative of some function?

I thought about this a lot and consulted a lot of people but everyone had contradicting answers. I am a high school student. please help.
1
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1answer
35 views

Differentiability of $\sum x^j$

Prove that $\sum x^j$ is differentiable on (-1,1), and $$\frac{d}{dx} \sum x^j = \sum (j+1) x^j$$ I am able to prove that $\sum x^j$ converges uniformly to $\frac{1}{1+x}$. However, how do I get this ...
2
votes
2answers
91 views

Find the derivative of $1/\sqrt{1+x^2-\cos^2x-e^{2\pi \cos(\sin 1/x)}}$

(calculus) How can I prove that $$\frac{d}{dx}\frac{1}{\sqrt{1+x^2-\cos^2x-e^{2\pi \cos(\sin 1/x)}}}=\frac{-\frac{\displaystyle\pi\sin(\sin(1/x))\cos(1/x)e^{2\pi\cos(\sin(1/x))}}{x^2}+x+\sin x+\cos ...
1
vote
1answer
46 views

Differentiability in two variables - directional derivative & gradient

I have read a chapter about differentiability in two variables. I now have two questions: Why do we need the constraint that $|\vec{u}|=1$ when we calculate the directional derivative? Definition of ...
3
votes
1answer
47 views

Differentiation of $u(t)=\int_0^t h(s,t)ds, \ \forall t \in \mathbb{R}$ with the multivariable chain rule

Problem: Let $h: \mathbb{R}^2 \to \mathbb{R}$ be continuous and differentiable with respect to its second variable, define $u(t)= \displaystyle \int_0^t h(s,t)ds, \ t \in \mathbb{R}$ In an ...
4
votes
1answer
88 views

Uncomfortable using Leibniz notation for the chain rule.

I am working through the following solved problem which uses separation of variables to get two ODEs. The problem is to show that $$\frac{1}{\sin\theta ...
0
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2answers
33 views

Stationary points of a function

$F(x)=x^3+Ax+B$ has a stationary point at $(-2,3)$. a) Find $A$ and $B$ and then find the nature of all stationary points. Thank you!
0
votes
1answer
84 views

Weierstrass Caratheodory on open interval

I have been working on this question for a while now, and if I have understood it correctly shouldn't the answer be that $\phi_{c}=f'(x)$ for all $x \in (a,b)$ as the function f , is now said to be ...
1
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0answers
32 views

Derivatives of function defined implicitly

The two equations $F(x,y,u,v)=0$ and $G(x,y,u,v)=0$ determine $x$ and $y$ implicitly as functions of $u$ and $v$, say $x=X(u,v)$ and $y=Y(u,v)$. Show that $$\frac{\partial X}{\partial ...
4
votes
1answer
139 views

Do rational and irrational numbers flip-flop?

I have found out that between every 2 rational numbers there is an irrational number, and between every 2 irrational numbers, there is a rational number. Does this mean that the rational and ...
3
votes
3answers
511 views

Finding a tangent to an ellipse parallel to a given line

Problem: Find the lines that are tangent to the ellipse $x^2 + 4y^2 = 8$ and parallel to $x +2y = 6$. I tried to find the derivative of $x^2 + 4y^2 = 8$ and I got: $$\frac{dx}{dy} = -\frac{x}{2y}.$$ ...