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

learn more… | top users | synonyms (2)

0
votes
2answers
31 views

derivative of $\frac{d}{dn}(1+\epsilon/2n)^n.$

I need to show that derivative of $\frac{d}{dn}(1+\frac{\epsilon}{2n})^n > 0.$ I use formula $(a^x)' = a^x\ln x.$ For now i have: $\frac{d}{dn}(1+\frac{\epsilon}{2n})^n = ...
2
votes
1answer
75 views

if $f(x)$ is differentiable at a x, prove that: $\lim_{h\to0}\frac{f(x+h)-f(x-h)}{2h}$

If $f(x)$ is differentiable at x, I need to prove that $\lim_{h\to0}\frac{f(x+h)-f(x-h)}{2h}$ exist and is finite. so if $f(x)$ is differentiable at a $x$, the difference quotient exist for this ...
0
votes
1answer
31 views

Optimisation problem - circle and square

A piece of wire of length $20$cm is cut into $2$ parts. the first part is bent into a circle of radius $r$ in cm, the second into a square of side length $s$ in cm. a) write down an expression for ...
2
votes
1answer
34 views

Does $g'$ need to be continuous for $g(x_0) = 0$, $g'(x_0) \neq 0$ to imply $g$ changes sign in a neighborhood of $x_0$

The following theorem holds: Theorem: Let $g:\mathcal{A} \rightarrow \mathbb{R}$ be differentiable and let $x_0 \in \mathcal{A} $. If $g(x_0)=0, \; g'(x_0)\neq 0$ then $g$ changes sign at a ...
1
vote
1answer
28 views

diferential equation system differential operators method

$x'-3x+2y=t$ $y'+2x=e^t$ it is asked to solve by the mentioned method $\Delta(D)=D(D-5)$ $\Delta_1=1-e^t$ $\Delta_2=-2t-2e^t$ $yD^3(D-5)(D-1)=0$ $xD^2(D-5)(D-1)=0$ When solving for the ...
0
votes
4answers
44 views

Derivative of a composition of functions

The problem is as follows: Find $g^\prime (2),$ given that $g(x) = f(x^2 + 2)$ and $f(e^x) = \log(\sqrt{x}).$ The answer turns out to be: $\displaystyle \frac{1}{3\log6}$ I tried to use the chain ...
6
votes
1answer
66 views

How derivative relates to roots of original function

Assume $f$ is differentiable on $\mathbb{R}$. Show that for any $ k \in \mathbb{R}$, $f' + kf$ has a root between any two distinct roots of $f$. I am completely stumped on this. What are some good ...
1
vote
1answer
35 views

About derivative of the inverse function

I think I misunderstand something about derivative of the inverse function. Say we are transforming from (x,y) to (r, $\theta$), this requires calculating $\frac{\partial x}{\partial r}$. $$x=rcos ...
1
vote
1answer
51 views

Partial derivative with matrices

I have reforumulated my problem of computing some quantities $\mathbf{a}\in R^{m}$ from $\mathbf{b}\in R^{n}$ in a matricial form: $$\mathbf{b} = (C\odot(\mathbf{1}_{n}\cdot \mathbf{a}^{T}))\cdot ...
3
votes
3answers
162 views

How to solve given expression?

We know that the derivative $f'(1)=3$. $$ \lim_{h \to 0} \frac{f(1-5h^2)-f(1+3h^2)}{h^2(h+1)}=? $$ I try to solve it by applying L'Hôpital's rule, but answer was incorrect. Since $f$ is ...
0
votes
1answer
41 views

Construct a complete 3rd order ODE with constants coefficients knowing 2 particular solutions and one particular solution of the homogeneous equation:

Construct a complete 3rd order ODE with constants coefficients knowing 2 particular solutions of this equation: $y_2=\ln(x)$ $y_1=x+\ln(x)$ and one particular solution of the homogeneous equation: ...
1
vote
2answers
37 views

reduction order method $xy''-(1+x)y'+y=x^2e^{2x}$, $y_1=1+x$

reduction order method $xy''-(1+x)y'+y=x^2e^{2x}$, being $y_1=1+x$ a solution of the homogeneous equation. I made y=u(1+x) and got $u''(x^2+x)-u'(x^2+1)=x^2e^{2x}$ Then i did $u'=w$ and obtained ...
2
votes
2answers
54 views

If a differentiable function approaches $-\infty$ as a limit from the positive side, must its derivative simultaneously approach $\infty$?

Can we say that if $g: (0, \infty)\rightarrow\Bbb{R}$ is a differentiable function and $\lim \limits_{x \to 0+}g(x)= -\infty$, then $\lim \limits_{x \to 0+}g'(x)= +\infty$ is always true? I ...
1
vote
1answer
82 views

Solving Partial Differential Equation $ \Delta u = 4 $

How can I solve this equation: $ \Delta u = 4 \\ u(x,x)=2x^2 \\ u_x(x,x) = 2x$ where $u=u(x,y)$ using substitution: $ \Phi ^{-1}(s,t) = (x-y,y) $? My attempt to solve this: $v=u \circ \Phi ...
3
votes
0answers
33 views

Axes-intersections of normal tangents to an ellipse

Question: What values can $x_T$,$y_T$,$x_N$, and $y_N$ take on? Let $T$ and $N$ be the tangent and normal lines to the ellipse $\frac{x^2}{9} + \frac{y^2}{4} = 1$ at any point on the ellipse in the ...
0
votes
2answers
65 views

if f'(x) is odd then f(x) is even?

Im trying to prove but every proof I encounter can also prove that if f'(x) is even then f(x) is odd and this is not correct (x^3 + 1 for example) thanks!
2
votes
1answer
42 views

reduction order method $(2-x)y'''+(2x-3)y''-xy'+y=0$, $y_1=e^x$

$(2-x)y'''+(2x-3)y''-xy'+y=0$, $x<2$ being $y_1=e^x$ a solution for the homogeneous equation. making $y=ue^x$ i came to $u''+u'''=0$ making $u''=w$ , $w'+w=0$ this way $w=e^{-x}*c_1$ and ...
0
votes
0answers
18 views

Clarifications about SDEs, Differentials & Derivatives

A general SDE look like the following: $$ \mathrm{d}\psi=a\mathop{}\!\mathrm{d}t+b\mathop{}\!\mathrm{d}W,\tag{1} $$ where $\psi:t\mapsto y = \psi(t)$ is the solution, while $a$ and $b$ can be both, ...
0
votes
2answers
37 views

About normal derivative

Let $w:\Omega \subset \mathbb{R}^n \longrightarrow \mathbb{R}$, where $\Omega$ is an open, bounded, connected set, $w \in C^2(\Omega)\cap C(\overline\Omega)$ and $x_0 \in \partial \Omega$ such that ...
1
vote
2answers
41 views

Using limit laws while calculating the difference quotient?

I'm trying to calculate the difference quotient of $f(x) = x|x|$ to calculate to derivative at $x=0$. Now when I try to do: $ \lim_{h\to0} f(x)=\frac{(x+h)|x+h|-x|x|}{h}$ it just seems too ...
0
votes
1answer
48 views

reduction of order method $xy''+2y'-xy=-e^x$ ,$y_1=\frac{e^x}{x}$

Reduction of order method $$xy''+2y'-xy=-e^x$$ being $$y_1=\frac{e^x}{x}$$ a solution of the associated homogeneous differential equation. I have followed the method and i came to ...
1
vote
1answer
38 views

Show that a given line is tangent to a whole family of curves

Here is my question: For each real parameter, consider the function defined as follows $$f_m=\sqrt{mx^2-2(m-1)x+m}.$$ Show that the line $y=\frac{\sqrt{2}}{2}(x+1)$ is tangent to each curve $C_m$, ...
2
votes
2answers
47 views

Plotting a function (by hand) if the second derivative is hard to find

In plotting graphics we use the first derivative to find critical points and in which intervals the function grows and becomes smaller. We can insert the critical points in the second derivative to ...
2
votes
1answer
48 views

Showing Differentiability of Function

Prove that $$f(t) = \int_{1}^{\infty} \frac{\sin(tx)}{1+x^{2}} dx $$ is differentiable on $(0, \infty)$. I tried to use dominated convergence theorem but have trouble finding the dominating ...
4
votes
2answers
118 views

(complex measures) $d\nu=d\lambda +f\,dm \Rightarrow d|\nu|=d|\lambda| +|f|\,dm$ and $f\in L^1(\nu)\Rightarrow f\in L^1(|\nu|)$

Two simple exercises on complex measures I don't know how to solve, from Folland's Real Analysis. I find it difficult to manipulate the definitions in computations. For context, if $\nu$ is a complex ...
1
vote
4answers
51 views

value of $\frac{d}{dx}f(x,f(x,x))$

$f(x,y)$ is differentiable at $(1,1)$, and $f(1,1)=\frac{\partial f}{\partial x}(1,1)=\frac{\partial f}{\partial y}(1,1)$, then what is the value of $\frac{d}{dx}f(x,f(x,x))$ when $x=1$? I applied ...
1
vote
0answers
52 views

Show $k$-form/chain identity

Let $\omega$ be a closed $k$-form on $\mathbb{R}^n$ and $c:I^k\rightarrow\mathbb{R}^n$ a $k$-cube on $\mathbb{R}^n$. Let $\mathbb{X}$ be a vector field on $\mathbb{R}^n$ with flow $\Phi_t$. Show that ...
0
votes
0answers
13 views

Analyze functions ($\exp(l) E_1(l)$ and $l\exp(l-1) E_1(l-1)$) that contain an exponential integral

Let $f_1(l)= \exp(l) E_1(l)$ and $f_2(l)= l\exp(l-1) E_1(l-1)$, where $E_1(.)$ is the exponential integral function. When I plot these 2 functions, I notice that $f_1$ and $f_2$ are 2 decreasing ...
0
votes
0answers
24 views

Partial derivative with small change in the input

Reviewing a book on the back propagation method for training a neural network, I am stuck on the calculus here, where the partial derivative of a function $C$ with respect to $z$, $\frac{\partial ...
3
votes
1answer
41 views

Can somebody help me on a simple chain rule differentiation problem [As level]

It's my first time using this forums so please let me know if I'm doing anything wrong and pardon for my stupid question. So I believe that when you differentiate $2$ functions e.g $y = f(g(x))$, you ...
4
votes
1answer
41 views

Indefinite integral which is not differentiable

I would like to know if there exists an indefinite integral which is not differentiable? Is this possible? That is, I want to know if there exists a real function F defined in a interval $[a,b]$ by ...
4
votes
0answers
48 views

For every $x \in [a,b] , \exists n_x\in \mathbb Z^+$ such that $f^{(n_x) }(x)=0$ ; then to prove $f$ is a polynomial in $[a,b]$

Let $f:[a,b] \to \mathbb R$ be a continuous function having derivatives of all order such that for every $x \in [a,b] , \exists n_x\in \mathbb Z^+$ such that $f^{(n_x) }(x)=0$ ; then how do I show ...
2
votes
3answers
60 views

Derivative of absolute value of $|x^5|$

Differentiate $|x^5|$. I know the formula for the derivative of absolute value but I can't seem to apply it to get $5x|x^3|$.
0
votes
1answer
32 views

Construction of mollifiers to generate a desire function

I am currently reading a book on analysis but do not understand the following: It claims that we can construct a function $\phi: \mathbb{R}^3 \rightarrow \mathbb{R}$ such that $1.$ $\quad $ $\phi ...
0
votes
3answers
88 views

Prove that a function is differentiable at a point

At which values of $x$ is $f(x)$ differentiable? $f(x) = \begin{cases} 1-e^{-x}, & \text{$x \gt 0$} \\ \ln(1-x), & \text{$x\le 0$} \end{cases}$ I first proved that $f(x)$ is continuous for ...
3
votes
3answers
49 views

Derivative, the tangent line $y = \sqrt{9-4x}$ at point (-4,5)

The function is given, $$y=\sqrt{9-4x}$$ Now I have to find the tangent line at point (-4,5). This is how i did it previously... Derivative; The Tangent line $y=x^2 -4x -5 ; (-2,7)$ Now, I think ...
0
votes
0answers
24 views

Find a complex-valued $g(u,v)$ such that $L_\mathbb{Y}g=img$

Let $F$ be a diffeomorphism between open $U$ and $V$ in $\mathbb{R}^n$. Let $\mathbb{Y}$ be a vector field on $V$ and $\omega$ a $k$-form on $V$. Given the identity ...
0
votes
4answers
74 views

Derivative of $e^{-x^2}$ [closed]

I've been doing pretty well with derivatives but I don't know how to find the derivative of the following function: $f(x) = e^{-x^2}$
1
vote
2answers
91 views

Proving differentiability

Prove that $$f(t) = \int_{0}^{1} e^{tx} x^{-1/3} dx $$ is differentiable at every $t \in \mathbb{R}$. ATTEMPT: I show $f'(t)$ exists at every $t \in \mathbb{R}$, i.e. the limit $$\lim_{h \to 0} ...
4
votes
1answer
72 views

$f:\mathbb R \to \mathbb R$ is a differentiable function such that $f'(x)\le r<1 $ , does $f$ necessarily have a fixed point ? [duplicate]

Let $f:\mathbb R \to \mathbb R$ be a differentiable function . If $\exists r \in \mathbb R $ such that $|f'(x)|\le r<1 , \forall x \in \mathbb R$ then using Lagrange's theorem one can show $f$ is a ...
2
votes
1answer
35 views

General Solution to Almost Riccati Like Equation

Consider the differential equation $$ y' = a_0(x) + a_1(x)y + a_2(x)\frac{1}{y}$$ I am attempting to find the general solution to this. One thing I can note is that the entire equation can be ...
0
votes
1answer
32 views

tetrahedron volume in the first octant

The surface is given: xyz = 2 It is in the first octant so x > 0, y > 0, z > 0. The tangent plane taken at any point of this surface binds with the coordinate axes to form a tetrahedron. Task: ...
3
votes
2answers
81 views

Faulty application of the Fundamental Theorem of Calculus to $f(x) = 0$ for $x\ne 0$, $f(0)=1$

I think I have given a fallacious proof but I can't seem to find what is wrong with it. Suppose $f : \mathbb{R} \rightarrow \mathbb{R}$ has the property that $\forall a,b \in \mathbb{R}. \int_a^b ...
1
vote
1answer
66 views

Show $D^2=0$ iff $D=e^{-f}de^{f}$ for some function $f$ , where $D\omega := d\omega+\alpha \wedge \omega $

Let $\alpha$ be a $1$-form on $\mathbb{R}^n$. Define the following which takes $k$-forms to $(k+1)$-forms. $$D\omega := d\omega+\alpha \wedge \omega $$ Show that $D^2=0$ iff $D=e^{-f}de^{f}$ for ...
2
votes
1answer
52 views

$\frac{f(x)-f(0)}{g(x)-g(0)}=\frac{f'(\nu(x))}{g'(\nu(x))} $ ,the value of the limit: $\lim_{x \to 0^+} \frac{\nu(x)}{x} $

Good evening, I thought a lot about this issue. I think I have to apply Lagrange, Taylor. Can someone help me to calculate this limit? $$f,g \in C^2 [0,1]: \\ f'(0)g''(0) \ne f''(0) g'(0) \\ ...
0
votes
0answers
17 views

Looking for a relationship between two push-forward maps

In a push-forward question I was to compute $(F_*\mathbb{X})(F(u,v))$, the next part is to calculate $(F_*\mathbb{X})(x,y)$. I have computed the first part and am able to compute the second part, ...
3
votes
1answer
30 views

Find the general solution of the simple harmonic oscillator

Question: Find the general solution of the simple harmonic oscillator equation, $$\ddot{x}=-\omega^2x$$ My answer: $x(t)=A\cos(\omega t)+B\sin(\omega t)$ Solution given: $x(t)=x_0\cos(\omega ...
1
vote
0answers
23 views

Could anyone help check the derivation?

I'm reading this paper and I'm trying to derive the gradient equation in (5) of the paper. But I couldn't get the right answer. My derivation is as follows. Could you help check it please?
0
votes
1answer
51 views

Understanding the definition of a pullback of a differential $k$-form and applying it in $1-d$

I am having trouble understanding the definition of a pullback of a differential k-form in a basic course in differentiable geometry. This is the definition I am given. I believe it is easier to ...
0
votes
0answers
25 views

Total derivative w.r.t. multiplied variables

I'm stuck in getting a Jacobian in Newton's method and need some help for the following derivative. Let $z=x\times y$. I want to get $\frac{\rm d f_1}{\rm d z}$ and $\frac{\rm d f_2}{\rm d z}$, where ...