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

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2
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1answer
30 views

Stationary Points with Logarithms?

I have this question: For the function $f(x)=x-2\ln(x^2+3)$: Find the two stationary points of this function, and enter them in the increasing order. I know how to find the stationary ...
0
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2answers
54 views

Matrix derivative of $X^TY^TYX$

I am trying to calculate the following derivative, involving $X$ and $Y$ matrices: $$ \frac{\partial}{\partial X}X^TY^TYX $$ I have tried something similar to the approach in Vector derivation of ...
0
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0answers
20 views

What value of t do I choose for the following Directional Derivative

Find the directional derivative of $f\left( x,y,z\right) =x^{2}+yz$ At $\left( 1,3,2)\right)$ in the direction of increasing t along the path : $r\left( t\right) =t^{2}i+3tj+\left( ...
3
votes
4answers
107 views

explanation of $ \frac{dy}{dx} = \frac{1}{\frac{dx}{dy}} $?

I'm studying about derivative of inverse function. The teacher in the video (https://www.youtube.com/watch?v=3ReOtNCYuBw) (at 9:00 minute) said this if a differentiable function, f has an inverse, ...
3
votes
2answers
69 views

Is there a function on a compact interval that is differentiable but not Lipschitz continuous?

Consider a function $f:[a,b]\rightarrow \mathbb{R}$, does there exist a differentiable function that is not Lipschitz continuous? After discussing this with friends we have come to the conclusion ...
1
vote
1answer
75 views

$\lim_{x\to 1/2} \left(\frac{\tan(\pi x)}{2x-1}+\frac{2}{\pi(2x-1)^2}\right)$ using L' Hopital

My task is to find $$\lim_{x\to 1/2} \left(\frac{\tan(\pi x)}{2x-1}+\frac{2}{\pi(2x-1)^2}\right)$$ using L' Hopital, so I first wrote it in a way that the numerator and denominator have the limit $0$ ...
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2answers
62 views

Differentiate $y=2(\ln x)^ \frac x2$

I tried doing the chain-rule but when I compared my answers to my TI-nspire and Wolfram I was missing a whole section. I have no idea how to differentiate $f(x)^{g(x)}$.
0
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0answers
32 views

Differential of a function with components

Assuming I have this function $$f(x, y, z) = 3x - 2xz^2 + e^{xy} + y^2$$ If I have to find its differential, then I will find that the differential is $$df = (3 - 2z^2 + ye^{xy}) dx + (xe^{xy} + ...
1
vote
1answer
46 views

Find local maximum or minimum in 2 variable function

So, I encountered a question (don't worry it's not H.W.) where I have a function with two variables, and I need to find local maximum / minimum points if exists. (More precisely, it is a utility ...
2
votes
1answer
65 views

Absolute continuity and Radon-Nikodym derivative

Let $\nu$ be a measure and $\mu$ a finite measure on $(X,\Sigma)$ with $\nu \ll \mu$. (All $\mu$-null sets are $\nu$-null.) Theorem: There exists a measurable $f:X \to [0,\infty]$ such that ...
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0answers
20 views

Critical points and the lie bracket

Let $f\in C^{\infty}(M,\mathbb R),p\in M$ and $(x,U)$ a chart of $M$ at $p$. The points $p$ is called critical point of $f$ if $$d(f\circ x^{-1})(x(p))=0.$$ Now I want to show that for a critical ...
6
votes
0answers
89 views

How should I calculate the $n$th derivative of this expression?

What would be the $n$th derivative of $ f (x) = x ^ x$ I have reached the fifth derivative, very long indeed but I see no pattern that will help me find a general expression. 1 D $y=x^x / ln ...
1
vote
1answer
24 views

Derivative of the inverse of exponential function a^x, with a>0 and a≠1

While studying exponential functions, I understood that $$\frac{d}{dx}a^x=(\ln a)a^x.$$ I also learned previously that if $g(x)$ is the inverse of $f(x)$, then the derivative of $g(x)$ and the ...
1
vote
3answers
33 views

Confusion regarding slope of a tangent to a parabola

I had learnt that differentiating the function $y=f(x)$ and putting the value of a point $(x_1,y_1)$ would give the slope of the tangent to the function at $(x_1,y_1)$. In other words, to find the ...
1
vote
3answers
53 views

Understanding meaning of $f''$ for $x^2$ and $x^4$

If $y=x^4$ Then $y' = 4x^3$ and $y'' = 12x^2$ At $x=0$, $y'=0$ and $y''=0$. So, at $x=0$, the gradient is zero is not increasing or decreasing at that point. I can believe this if I look at a plot ...
0
votes
0answers
25 views

How do I find $r(x,x_0)$ of function?

$$r(x_0, x) := \frac{f(x) − f(x_0)}{x − x_0} − f′(x_0)$$ We're given $$f(x)=x^3-4x-2\\ x_0=2$$ I have found $f′(x)=3x^2-4$ and $f′(x_0)=8$. But still I can't get $r(x_0, x)$. If someone could help ...
2
votes
2answers
72 views

Finding the Fourth Derivative

Let $f(x)$ be a four-times differential function such that $f(2x^2-1)=2xf(x)$ What is the value of $f''''(0)$? A brute force approach is differentiating the given condition 4 times and finding ...
0
votes
2answers
21 views

Find the maximum value of $y$ with differentiation

I have the following problem to solve but I don't know where to start. The sum of two positive numbers is $50$. One number is $x$ so the other number must be $50 − x$. Let the product be $y$. Use ...
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2answers
78 views

Calculate the second-order derivative.

Let $x \in \mathbb{R}^m$ be our variable. I would like to know what is: $$ \frac{\partial^2 \text{Tr}\big((A+B^\text{T}\textbf{diag}(x)B)^{-1}\big)}{\partial x_i \partial x_j}. $$ $A \in ...
2
votes
1answer
59 views

Saddle points and the second derivative changing sign

Let $f$ be a some real-valued function differentiable at least three times (I want to say twice, or maybe twice with a continuous second derivative, but I'll play it safe). A saddle point of $f$ is ...
0
votes
1answer
31 views

Analysing functions

Let $f(x)$ be a twice differentiable function on $[1,3]$ and $f(1) = f(3)$. $|f''(x)|<2$, for all $x$ in $[1,3]$. Then in $[1,3]$, what is the range of $|f'(x)|$? The first approach to this ...
3
votes
1answer
79 views

On the differentiability of monotone functions

It is well known that if $f$ is monotone on $[a,b]$, then $f$ is differentiable almost everywhere on $[a,b]$. I am trying to find a condition which forces $f$ to be differentiable at its endpoints ...
1
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2answers
76 views

A circle centered at the origin is tangent to $y=2^x$. What is the radius of the circle?

I feel as though I am doing the analytical part correctly, however, where I am facing the roadblocks in this problem is in the actual algebra itself. Perhaps I am not doing something right in my ...
0
votes
1answer
13 views

Classifying points found with derivative

I've been trying to learn derivatives recently. I have troubles with finding local minimums and maximums given an equation I have understood this far $y=2x^3-9x^2+12x-5$ $dy/dx= 6x^2-18x+12$ $x=1$ ...
0
votes
1answer
26 views

Smoothness of division of infinitely differentiable functions

Suppose I have a $C^\infty$ function $f\colon\mathbb R\to\mathbb R$, $f(0)=0$, is it true that $g(x)=\frac{f(x)}{x}$ is also a $C^\infty$? If it is true, how do I prove it? Generalized to ...
0
votes
1answer
37 views

differntiation of functions [closed]

Let $f,g$ and $h$ be 3 differential functions. Given that $f(0)=1,g(0)=2 ,h(0)=3$ and $(fg)' (0)=6,(gh)' (0)=4$ and$(fh)' (0)=5$ then find the value of $(fgh)' (0)$. I am not able to understand how ...
2
votes
3answers
221 views

Related Rates - Distance between two ships

At noon, a vessel is sailing due north at the uniform rate of $15$ kilometers per hour. Another vessel, $30$ km due north of the first vessel, is sailing due east at the uniform rate of $20$ ...
0
votes
1answer
35 views

Bound for function with constant /periodic second derivative

Consider a function $f : \mathbb{R}\to\mathbb{R}$ with $f''$ continuous and $f''(x)=f''(x+1)$ for all real numbers x. I need to show that there exists a real positive number $c$ such that $f(x)\leq ...
1
vote
1answer
32 views

Some doubts regarding local minimums of twice-differentiable functions

(Sorry if this question, along with some other recent questions, is trivial or there's errors in how I worded it. I'm a beginner calculus student) Let $f:[a,b] \rightarrow \mathbb{R}$ be a continuous ...
2
votes
1answer
38 views

Calculate the value $-te^{-t} - e^{-t}$ at t $\rightarrow\infty$

I am trying to evaluate the following equation at $t \rightarrow \infty$ and $t \rightarrow 0$: $-te^{-t} - e^{-t}$ I am trying to use L'Hopital's Rule to evaluate $-te^{-t}$ at $\infty$, so it ...
1
vote
3answers
60 views

Simple explanation of the differentiation of $\ln(f(x))$

Could somebody explain why the derivative of $\ln[f(x)]$ = $f'(x)/f(x)$ . Why is it not simply $1/f(x)$ as is the case for the derivative of $\ln(x)$ being $1/x$?
1
vote
0answers
44 views

Derivative of the upper incomplete gamma function.

I wish to compute the derivative of the upper incomplete gamma function \begin{equation} \Gamma(s,x) = \int_{x}^\infty t^{s-1}e^{-t} \, dt . \end{equation} Wikipedia states of the derivative of ...
1
vote
1answer
25 views

Can there be a function, that with the slope $f'(x_0) = 0$ at the single root $x_0$

Let $f$ be a function of the form $$ f(x) = \sum_{i = 0}^n a_i x^i = a_nx^n + a_{n-1}x^{n-1} + ... + a_1x+ a_0 $$ with $a_n, a_{n - 1}, ..., a_1,a_0 \in \mathbb{R}$. Let $x_o$ be a single root of $f$ ...
0
votes
2answers
32 views

Linear derivative of nonlinear odd function possible?

Can the derivative of an odd , non-linear function ever be linear? Need a mathematical proof or example of if being linear. Thanks
0
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1answer
60 views

How do I deal with this Diag operator when differentiating with respect to a matrix?

This question is an extension of this question. The arbitrary function $B(\cdot)$ is now as specified as follows: $$ B(\mu, \sigma^{2}) = \exp \left(\mu + \frac{1}{2}\sigma^{2} \right). $$ So if I ...
2
votes
2answers
34 views

Approximation of $2$nd Derivative Up to $O(h^4)$

Investigate if it is possible to obtain 4th order accuracy using 5 points for a 2nd derivative approximation, i.e. is it possible to determine a, b, c, d, e in $$y''(0) = ...
0
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0answers
54 views

Left hand and right hand derivatives of $x\sin ({1\over x})$ at $0$

If $$f(x)= \begin{cases} x\sin\left(\frac{1}{x}\right) & x\neq 0 \\0 & x=0 \end{cases}$$ Then $A.$ $f$ is continuous. $B.$ $f'(0+)$ exists but $f'(0-)$ does not. ...
2
votes
1answer
42 views

Partial derivative of $f(x,y)=g(2x+5y)$

Let $g:\mathbb{R}\to\mathbb{R}$ be a differentiable function of one variable and let $f(x,y)=g(2x+5y)$. How do I find the partial derivative $f_x(x,y)$? I am more interested in an explanation than ...
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vote
3answers
47 views

Sufficient condition to show $f$ is monotonically increasing in some neighborhood

I am curious if the following statement holds. Let $f:[a,b] \rightarrow \mathbb{R}$ be a continuous function differentiable on the open interval $(a,b)$. Then if $f'(c)>0$ for some $c \in (a,b)$, ...
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0answers
17 views

Ordinal Hinge Loss Derivative?

According to This paper, the ordinal hinge loss can be defined as follows in (4). I believe that 'a' is the true value and 'u' is the predicted one. My question is if it is possible to re-write this ...
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0answers
32 views

Beginner deravative of peice wise question and connection with differentiation and continouity

Hi I am wondering if someone can help to explain to me the following; Say we have $$f(x)= \begin{cases} x^{2}, &\text{ if $x \ge 0$ } \\ 0, &\text{if $x \lt 0$} \\ \end{cases}$$ and we want ...
1
vote
1answer
82 views

derivative of expected value with respect to parameter in both pdf and expectation

Say $X \sim N(\mu, \sigma^2)$ with pdf $f(x, \mu)$. We are interested in expectation of $g(x)$. Then $$E[g(x, \mu)] = \int_{-\infty}^{\infty} g(x, \mu) f(x, \mu) dx$$ Now I want partial derivative of ...
0
votes
0answers
25 views

Polar co-ordinates dr/dtheta

How can you visualise what is the curve doing by calculating Dr/dtheta in polar co-ordinates form. Also, what will it mean for Dr/dtheta to be zero? Thank you.
2
votes
1answer
40 views

PDE - $y^2 \frac{\partial^2 u}{\partial x^2}=x^2 \frac{\partial^2 u}{\partial y^2}$ - how to derive the general solution

$\mathbf{y^2 \frac{\partial^2 u}{\partial x^2}=x^2 \frac{\partial^2 u}{\partial y^2}}$ is a hyperbolic PDE where $\xi =y^2+x^2$ $\eta =y^2-x^2$ which gives ...
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0answers
41 views

What is the partial derivatives of this log map with respect to u and v

Given function $Log_u(x) = \frac{arccos(u^Tx)}{\sqrt{1 - (u^Tx)^2}}(1 - uu^T)x$, where $u$ and $x$ are vectors in $\mathbb{R}^n$ and $$||u||= ||x|| = 1$$. How to calculate $\frac{\partial ...
0
votes
1answer
20 views

Find area of triangle which sides is limited by two functions and the x axis

I'm studying for my math exam and I'm stuck on the following question "A triangle is limited by the x axis and the two functions $y=kx$ och $y=\frac{1}{k}x+k$ where k > 1. Determine the smalest ...
0
votes
2answers
53 views

$f'(x) = \sqrt{1-f(x)^2}$, then $(f^{-1})'(x) =$

Math StackExchange, long time reader, first time writer. I have a question on inverse differentiation. The question is: Suppose $f'(x) = \sqrt{1-f(x)^2}$, then $(f^{-1})' (x) = ?$ I had a similar ...
0
votes
1answer
41 views

How to differentiate Residual sum of square

Residual sum of square (RSS) is defined as RSS(beta) = $(y-X * beta)^t (y-X * beta)X$ While differentiating RSS(beta) w.r.t to beta to find the minimum value of the function, author reaches the ...
0
votes
3answers
94 views

Find $g''(\pi/3)$ if given two definite integrals

Find $g''(\pi/3)$ if $$f(x) = \int_0^{\cos x} \sqrt{1+t^2} \, dt \text{ and } g(y) = \int_e^y f(x) \, dx.$$ So I realize I need to find the second derivative of $g(y)$, and the first step is to ...
3
votes
1answer
57 views

Closed form for $n$-th derivative of $\sqrt{f(x)}$ for general $f(x)$

Let's assume we have an inifinitely differentiable real valued function $f(x)$, and we have a closed form expression for all its derivatives. Is it then possible to find a closed form for the $n$-th ...