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

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1
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3answers
28 views

Limit of derivative does not exist, while limit of difference quotient is infinite

Can anyone show an example of a function $f$ of a real variabile such that $f$ is differentiable on a neighborhood of a point $x_0 \in \mathbb{R}$, except at $x_0$ itself; $f$ is continuous at ...
0
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0answers
8 views

Derivative with respect to vectors related through a matrix

Consider a function $g: \mathbb{R}^r \to \mathbb{R} $ and two vectors $\mathbf{b} \in \mathbb{R}^r$ and $\mathbf{c} \in \mathbb{R}^m$ such that $\mathbf{c} = \mathbf{A}\mathbf{b}$. If I calculate the ...
2
votes
2answers
59 views

Can $f''(x)$ exist if $f'(x)$ is undefined?

For example, the piecewise function $ f(x) = \begin{cases} \sqrt{1 - (x + 1)^2} &-2 \leq x \leq 0 \\ -\sqrt{1 - (x - 1)^2} &0 \leq x \leq 2 \end{cases} $ will, at $f(0)$, give $f'(0) = $ ...
0
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0answers
10 views

Derivative of equation in matrix form

I need to compute first derivatives of the following function $S(w)$ with respect to $w$. Then solve it. The reason behind that is to minimize $S(w)$. $S(w)=\sum_{i=1}^{n} w_i^{1/2} \bigg(y_i - ...
0
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2answers
17 views

Finding common tangent line to two functions

Sometimes you want to find the common tangent line of two functions. The first thing that comes to mind to a person that is learning basic calculus is that you should equal the derivatives of those ...
0
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1answer
27 views

Multivariable Calculus, Parametrization and extreme values

I want to find the extreme values of the function $f(x,y,z) = 2x + 2y + z$ under the constraints $x^2+y^2+z^2 \le 2$ and $x^2 + y^2 \le z$ The task is to use a parametrization of the two ...
2
votes
2answers
51 views

What is the partial derivative of $f(x,y(x))$?

What is the total derivative of $f(x,y(x,z))$ with respect to $x$? Is it $$\frac{\partial f}{\partial x}+\frac{\partial f}{\partial y}\frac{\partial y}{\partial x}?$$ If this is correct, what is ...
1
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2answers
57 views

Why is the function continuous at a point which gives the case 0/0?

I have this function : $f(x) = \frac{6x^2+18x+12}{x^2-4}$, the domain is R. How come its graph is continuous at $x = -2$? I know it can be simplified to $\frac{6(x+1)}{x-2}$ ( firstly $f(x) = ...
2
votes
1answer
34 views

How to compute $[\dot c, X]$ on a manifold?

Consider a smooth curve $c : [0,1] \to M$ and $X \in \mathcal X (M)$. How can one obtain an explicit formula for $[\dot c, X]$? I know the theoretical approach: for every $t \in [0,1]$ there exist a ...
1
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0answers
341 views

What is the upper-bound for this?

I am looking at a paper and am trying to understand how this bound was driven. The first part is clear, but not sure how you can extend it to the second part. So here is the first part: Assume ...
1
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2answers
93 views

How to differentiate this integral?

Given $$g(x)=\int_{0}^{x} (x-t)e^{t}dt$$ find out $g''(x)$ I thought of using Lebnitz theorem to differentiate it but using Lebnitz I get this $g'(x)=1\cdot (x-x)e^{x}=0$ I don't know how to find ...
0
votes
2answers
24 views

What is the instantaneous rate of change in the real world?

I can't grasp this concept of an instantaneous change of rate. How could a point on a function graph have a rate of change in the first place? In this moment I just know that it is named the ...
11
votes
2answers
58 views

Solution to $\frac{d}{d\frac{1}{x}} x$

If I want to solve $$\frac{d}{d\frac{1}{x}} x$$ is my approach correct? As $$\begin{align*} \frac{d}{d\frac{1}{x}}x&=\\ \text{with }\frac{1}{x}&=y\\ ...
2
votes
0answers
31 views

How to prove that the following function has a unique mode?

I am trying to prove that the function $$f(\alpha)=n\ln \alpha-n\ln\Big(\sum_{i=1}^{n}t_i^\alpha+\int_{a}^{b}x^{\alpha+\beta-1}e^{-\lambda x^\beta}\,dx\Big)+(\alpha-1)\sum_{i=1}^{n}\ln t_i,$$ where ...
0
votes
0answers
41 views

Prove that the following function is convex?

I am trying to prove that the function $$g(\alpha)=\ln\Big(\sum_{i=1}^{n}t_i^\alpha+A(\alpha)\Big) ~~t_i, \alpha>0,$$ where $A(\alpha)=\int_{a}^{b}x^{\alpha+\beta-1}e^{-\lambda x^\beta}\,dx$,is ...
0
votes
1answer
26 views

Limit proof of derivative function

$f:[0,+\infty) \rightarrow \mathbb{R}$ twice differentiable. If $f''$is limited and there is $\lim\limits_{x\to \infty}f(x)$, show that $\lim\limits_{x\to \infty}f'(x) = 0$.
0
votes
0answers
7 views

Partial derivative of polynomial dependant on previous time values

I have not touched calculus for a few years, and I am not sure what is going on here. Any help would be greatly appreciated :) Essentially, let $p_{t} = \log p(y_{t}|h_{t},h_{t+1})$, where ...
1
vote
1answer
21 views

Function whose $n$-th derivative at $x=0$ is $n^3$ / evaluating the power series $\sum_{n=1}^\infty n^3\frac{x^n}{n!}$

I have troubles proving that the power series $\sum_{n=1}^\infty n^3\frac{x^n}{n!}$ represents the function $f(x)=e^x(x^3+3x^2+x)$. My idea was using the identity theorem for power series and the ...
0
votes
2answers
27 views

Confusing result obtained taking second derivative of ye^y

I was doing my calculus homework, and one of the questions asked for the first and second derivative of $ye^y=x$, I did the computations and arrived at $-(x+1)^{-2}$, which was a lot neater and ...
4
votes
2answers
42 views

Prove that $\lim\limits_{x\to\infty} f'(x)=0$

Let $f$ be a function in $(0,\infty)$ such that $f'(x)$ exists. In addition, $\lim\limits_{x\to \infty} f'(x)=L$ (finite) and $f(n)=0$ for every $n \in \Bbb N$. Prove that ...
0
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1answer
16 views

Lagrangian derivative quotient rule?

How do I find the Lagrangian derivative of: $$\frac{D}{Dt} \left(\frac{x}{y^{a}}\right) = 0$$ where a is a constant?
0
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2answers
31 views

Using definition of derivative to prove that $\lim_{h \to 0} \dfrac{f(x_0+bh)-f(x_0-ch)}{(b+c)h}=f'(x_0) $

I'm studying in preparation for a Mathematical Analysis I examination and I'm solving past exam exercises. If it's any indicator of difficulty, the exercise is Exercise 4 of 4, part $a$ and graded for ...
1
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4answers
66 views

Minimum distance between the curves $f(x) =e^x$ and $g(x) =\ln x$ [on hold]

What is the minimum distance between the curves $f(x) =e^x$ and $g(x) = \ln x$? I didn't understand how to solve the problem. Please help me.
0
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0answers
27 views

Fourier transform calculus of tempered distributions

For example I wanted to ask confirmation of this calculation, if $u \in \mathcal{S}'(\mathbb{R}^n)$ then $\widehat{D^\alpha u} =(2\pi i \xi)^\alpha \widehat{u}$. By definition $\langle \varphi , u ...
0
votes
1answer
12 views

Finding the rate at which the space diagonal of a cube is decreasing

This is the context of the question: I'm assuming by the space diagonal (although I'm not sure) to be the area of the right-angled traingle created by the diagonal. Let this space be $V$, then we ...
2
votes
3answers
17 views

Finding the formula for acceleration from $v=2s^3+5s$, where $s$ is the displacement at time $t$

This is the question: I first found $\frac{dv}{ds}=6s^2+5$, then I tried to find $\frac{ds}{dt}$ by messing about a little with implicit differentiation, but I had no luck and I therefore couldn't ...
0
votes
0answers
25 views

Range of Derivative

Let $g(x) = f(x)/(x+1)$, where $f(x)$ is differentiable on $x\in[0,5]$, such that $f(0)=4$ and $f(5)=-1$. What is the range of values $g'(c)$ for a $c$ belonging to $[0,5]$? Considering values of ...
1
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3answers
46 views

Why are these equations equal to a constant?

I am reading this part of a research paper where the author states that the left hand side of equations (12) and (13) must be equal to a constant. However I could not understand the explanation he ...
1
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1answer
19 views

Differentiable any finite number of times

Does there exist a pathological function which is differentiable any finite number of times as one wishes, but is not differentiable an infinite amount of times? Is it reasonable for such function to ...
0
votes
2answers
41 views

Is $f( x + iy) = e^{-x} e^{-iy}$ complex - differentiable?

I started by letting $u(x,y) = e^{-x}$ and $v(x,y) = e^{-iy}$ . I then tried to use the cauchy reiman equations : $\frac{\partial u}{\partial x} = \frac{\partial v}{\partial y}$ and $\frac{\partial ...
3
votes
1answer
28 views

Algebra of Linear differential operators, question on Commutativity and Association

The following is a discussion on the following second differential equation $$ \frac{dy^2}{dx} - y = 0 $$ So, let us introduce the following, convention and definition, represent the derivative ...
1
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2answers
53 views

Memorizing Formulas for Differentiation

Once upon a time, I memorized the following formula out of laziness. Let $k(x)=\frac{f(x)^{g(x)}h(x)+i(x)}{j(x)}$. Then $k'(x)$ is as follows. ...
0
votes
1answer
18 views

Using Rodrigues' formula to show a result

use the formula $P_n(x) = \dfrac{1}{2^nn!}\dfrac{d^n}{dx^n}((x^2-1)^n)$ to show that $P_{2n}(0) = \dfrac{(-1)^n(2n)!}{4^n(n!)^2}$ and odd terms are 0. I first subbed in 2n to the formula and got ...
-2
votes
1answer
22 views

Does the momentum of a particle depend on its position [on hold]

By definition: $\displaystyle \dot{x}=\frac{p}{m}$, where $p$ is the momentum of the particle $m$ is the mass, and $\dot{x}$ the velocity. As the velocity depends on the position of the particle(?) ...
0
votes
2answers
39 views

Is velocity a function of displacemnt?

The velocity $\displaystyle\vec{v}$ of a particle $=\frac{d\vec{x}}{dt}$. So surely this means that $\vec{v}$ is dependent on the position of the particle?
0
votes
1answer
26 views

Bound the variation of this decreasing function

Let $f(x)$ be a decreasing function defined over the interval $[0,a]$, with $f(0)=b$. The first derivative of $f(x)$, which is negative, is such that $f^\prime(x) > g(x)$, or equivalently ...
0
votes
0answers
31 views

Leibniz integral rule definition

https://en.wikipedia.org/wiki/Leibniz_integral_rule If we have an integral $$\int_{y_0}^{y_1} f(x, y) \,\mathrm{d}y$$ then for $x$ in $(x_0, x_1)$ the derivative of this integral is thus ...
0
votes
1answer
30 views

Proof of derivatives though first principle method

In school I was taught to memorize this formula. $ \frac {d} {dx} (\cos x) = - \sin x $ However, recently I found out a proof using the first principle (under the "Derivatives" chapter), but could ...
0
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3answers
25 views

Why does $\frac{1}{6e^{2y}}=\frac{1}{2x-8}$ in this context?

This is the context: I tried substituting $y=3e^{2x}+4$ into $6e^{2y}$but I wasn't able to go any further. Does anyone what exactly is being done in the last step?
0
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1answer
45 views

Points of contacts of tangents of the curve $y=\sin x$

Prove that the points of contacts of tangents of the curve $y={\sin x}$ drawn from origin lie on the curve $\frac{1}{x^2} - \frac{1}{y^2} = -1$
21
votes
2answers
692 views

Function that is the sum of all of its derivatives

I have just started learning about differential equations, as a result I started to think about this question but couldn't get anywhere. So I googled and wasn't able to find any particularly helpful ...
1
vote
1answer
12 views

$\frac{d}{dy} F(g(y),y) = ?$

Given that we know if the integral of $f(y)$ is $F(y)$ then we can say that $\frac{d}{dy} F(y) = F'(y) = f(y)$. But what does $\frac{d}{dy} F(g(y),y)$ equal to? Can we say that it is $f(g(y),y)?$
-1
votes
2answers
24 views

Need help in verifying if I am taking the derivative of $f(x) = \frac{x}{\cos(x)}$ correctly

I need to take the derivative of $f(x) = \frac{x}{\cos(x)}$. What I am doing: $$f'(x) = \frac{d\ (x\cos(x)^{-1})}{d \ x} + (\frac{d\ (x\cos(x)^{-1})}{d\ (\cos(x))} * \frac{d\ \cos(x)}{d\ x})$$ ...
0
votes
1answer
34 views

If $f$ is differentible at a point $x \in [a,b]$, then $f$ is continuous at $x$.

Proof. As $t\rightarrow x$, we have, by Theorem 4.4, (Baby Rudin, p.104) $f\left( t\right) -f\left( x\right)$ = $\dfrac {f\left( t\right) -f\left( x\right) } {t-x}\cdot \left( t-x\right) \rightarrow ...
0
votes
1answer
24 views

Differentiating a function composition

Given $g:R^n \rightarrow R^k$ and $h:R^k \rightarrow R$, we have $f(x) = h(g(x))$. Using the chain rule, we can differentiate $f(x)$ to get $f'(x) = \nabla^Th(g(x))g'(x)$ My question is why do we ...
0
votes
0answers
17 views

Relation between sub-tangent and sub-normal of $y^2 = (x+a)^3$

If the relation between sub-tangent and sub-tangent at any point on the curve $y^2 = (x+a)^3$ is $p(SN)=q(ST)$ then find the value of $p$ and $q$ where $SN$ is length of sub-normal and $ST$ is length ...
1
vote
4answers
73 views

Proving that the exponential inequality $e^x \ge x^e$ holds for all $x \ge 0$ [duplicate]

How does one prove that $$e^x \ge x^e$$ for all $x \ge 0$? I tried to do this by setting $f(x)=e^x-x^e$ Plotting this function shows this easily, as seen here. However, when I tried to prove ...
0
votes
0answers
45 views

Bound the first derivative of the following function: $f(x)=g(x)+h(x)$

Consider a decreasing function $f(x)$, resulting from the sum of two other decreasing functions: $f(x)=g(x)+h(x)$. All these 3 functions are positive. In addition, we have $g(0)=h(0)=1$. Further, we ...
1
vote
2answers
45 views

What does third derivative tell about inflection point?

I was trying to find the nature (maxima,minima,inflection points) of the function $$\frac{x^5}{20}-\frac{x^4}{12}+5=0$$ But I faced a conceptual problem.It is given in the solution to the problem ...
8
votes
1answer
73 views

Find all $f:\mathbb {R} \rightarrow \mathbb {R}$ where $f(f(x))=f'(x)f(x)+c$

Recently, while studying calculus, I have come across multiples problems which asked the following: If $f(x)$ is a polynomial, find all $f(x)$ that $f(f(x))=f'(x)f(x)+c$, where $c$ is a constant. ...