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

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3
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1answer
72 views

Evaluating an integral and differentiation

I'm trying to understand the math in a journal paper, but I'm stuck on figuring out one of the integrals. Here is the paper called, "Simultaneous optimization of the material properties and the ...
0
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0answers
25 views

higher order derivatives of three composite functions

How can I obtain a formula for higher order derivatives for composite of three functions as $f(g(h(x)))$?
1
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1answer
8 views

Need help with Application of Leibniz's Integral Rule

I have a result I am trying to understand: $\frac{\partial}{\partial \beta} \int_0^\beta b dF(b)^N = \beta N F(\beta)^{N-1} f(\beta)$ This is how I would like to think about the problem. The PDF of ...
0
votes
2answers
44 views

Chain rule for linear equations (Derivatives)

I am having a hard time understanding why the chain rule works. When going over a theorm, or feature of the maths in general, one starts of with the easiest examples to get to grips with said concept. ...
-1
votes
0answers
31 views

Derivative of Norm of a difference of vectors

I have an expression: $||ax-b||^{2}_{2}$ where both $x$ and $y$ are vectors. I want to find $\frac{d}{dx}||ax-b||^{2}_{2}$, which is the vector derivative of the norm wrt to the vector x. Does ...
1
vote
2answers
39 views

The derivative of a recurrence relation of functions

I am unsure of how to take the derivative of a recurrence relation of functions. For example consider the following recurrence relation: \begin{equation} \left\{ \begin{array}{cl} f_n(x) &= ...
-1
votes
1answer
33 views

Unboundedness of differential operator on test function

Currently I am studying the differential operator $T: L^2((0,1)) \to L^2((0,1))$ with the domain $D(T) = C_0^\infty((0,1))$. I am having difficulties finding a sequence to show the unboundedness of ...
1
vote
1answer
47 views

Which “approximate” value of f(0.98) is this question looking for?

In a section of a calculus workbook dealing with local linearity and linear approximations of functions, the following question is posed: Consider the function f(x) = aln(x+2). Given that f'(1) = ...
4
votes
2answers
57 views

Differential at a point and differential (Differential Geometry)

Given $f\in C^\infty(U)$, $U$ open set of $\mathbb{R}^n$, we define the differential of $f$ at $p$ $$ (df)_p:T_p\mathbb{R}^n\to\mathbb{R}\\ (df)_p(v):=v(f) $$ and the differential of $f$ $$ df:U\to ...
0
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0answers
25 views

Approximate solutions to differential equations

If one has a differential equation for $y(x)$. If this differential equation has two solutions one for $x\ll a$ and the other for $a\ll x$, where $a$ is constant real value. My question is at what ...
0
votes
1answer
77 views

The name of the theorem expressing the derivative of an integral with variable limits

What is the name of this theorem, or how to prove this? $f(x)=\displaystyle\frac{1}{\Delta x}\int_{x-\frac{\Delta x}{2}}^{x+\frac{\Delta x}{2}}h(\xi)d\xi$ $\Longrightarrow$ ...
1
vote
4answers
84 views

Why should I use derivatives and calculus?

I know that this question maybe sounds pretty generic, but it's a curiosity that I have and I didn't found any answer yet. I recently started studying calculus using this material where is said that ...
0
votes
2answers
27 views

Minimizing cost for a given volume

288 m3 tank will be made in the form of a rectangular prism. The cost of 1 m2 of top and bottom walls is 40 euros. The cost of 1 m2 of side wall is 30 euros. What should be the edges to be cheap as ...
0
votes
2answers
27 views

Derivative of dot product?

What's the derivative ${\partial \over \partial x} \langle x, f(x)\rangle$? According to the product rule it should be $1\cdot f(x) + x \cdot f'(x) $ but in my previous post I was told that this ...
0
votes
1answer
26 views

Second derivative numerical estimate - stability and approach

I would like to know how to estimate second derivatives of a function sampled discretely with constant spacing. Let there be a function $f(x)$. I sample its values $\{f(x_i)\}$ at points $\{x_i\}$ ...
-4
votes
3answers
64 views

What is the first derivative of $1\over x$ [closed]

I want to find out the first derivative of $1\over x$ but I'm not sure how. Can someone provide detailed explanation? Thank you.
8
votes
6answers
146 views

To show that $e^x > 1+x$ for any $x\ne 0$ [duplicate]

$$e^x>1+x$$ is what I want to show. So let's define a function: $$h\left(x\right)=e^x-x-1$$ and investigate its derivative: $$h'\left(x\right)=e^x-1$$. Easy to see that at $x=0$ it has a ...
0
votes
1answer
54 views

2 exercises: finding the limit and showing continuity and differentiability

part 1: $$\lim _{x\to _{x\to \frac{\pi }{2}^{-\:\:}} }\left(tg\left(x\right)\right)^{\sin\left(2x\right)}$$ so if $$\lim _{x\to _{x\to \frac{\pi }{2}^{-\:\:}} ...
0
votes
1answer
33 views

Speed of light moving on a wall.

While studying, I came upon this word problem: "A police car is 20 feet away from a long straight wall. Its beacon, rotating 1 revolution per second, shines a beam of light on the wall. How fast is ...
0
votes
2answers
27 views

Logarithmic derivative of Polygamma functions

While studying Gamma function and related functions I noticed that its logarithmic derivative (the so-called Digamma function) is studied more than its "normal" derivative but on the other hand I ...
0
votes
0answers
12 views

Derivative of implicit function - possible to bring in specific form?

Let $f(\alpha) := \sum_{j=0}^{N-1}\alpha^j = \frac{1-\alpha^N}{1-\alpha}$. I am analyzing an implicit equation of the form $g(v,\alpha) := f(\alpha) - \frac{c}{v} = 0$, where $c$ is a positive ...
1
vote
2answers
31 views

Show that it's image is R and prove that it is an injective and find the tangent of the opposite function

Q1.p1: To show that this function is injective and that it`s imgae is R. $$f\left(x\right)=x^3+3x+1$$ My solution: let's look at it's derivative: $f'\left(x\right)=3x^2+3\:>\:0$ and that's why it ...
0
votes
1answer
33 views

Show that the function $f(x,y) = |xy|$ is differentiable at 0, but is not of class $C^1$ in any neighborhood of 0.

The problem from Munkres' *Analysis on Manifold is that Show that the function $f(x,y) = |xy|$ is differentiable at 0, but is not of class $C^1$ in any neighborhood of 0. My thought on the first ...
2
votes
0answers
23 views

How to differentiate a harmonic function presented by Poisson integral formula

Let $h(x+iy)$ be a harmonic function in the open neighbourhood of the closed unit disc $\overline\Delta(0;1)$ of $\mathbb{C}.$ Then it can be presented by Poisson integral formula in the following ...
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2answers
24 views

Finding first and second derivative of an function with an absolute value

Given the equation $f(x)= |x^2-9|$ where $-4\le x\le 5$, I must find the extremes, as well as the concavities. This I know how to do. The issue is I'm unfamiliar on how to find the first and second ...
2
votes
2answers
44 views

Matrix Differentiation

Consider a differentiable function $f: \mathbb R \to \mathbb R$ and two $p\times 1$ vectors $x$ and $\theta$. Then define a new function as follows. $$ f\left( x^T\theta \right)x. $$ Now we want to ...
0
votes
2answers
34 views

Taking derivative of function $g: \mathbb{R} \to \mathbb{R}$ defined in terms of $f: \mathbb{R}^{n+1} \to \mathbb{R}$.

Suppose we are given $g(r): \mathbb{R} \to \mathbb{R}$ where $g(r) = f(ry, r^2s)$ for $f: \mathbb{R}^{n+1} \to \mathbb{R}$ where $y \in \mathbb{R}^n, s \in \mathbb{R}$. How do we determine ...
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vote
4answers
84 views

How to use definition of limit to compute the derivative of |x|

Using definition of limit, I need to show $$\lim_{\epsilon \to 0} \frac {|x + \epsilon| - |x|}{\epsilon} = \frac {x}{|x| }, x \neq 0$$ How should I proceed to get out of the absolute value signs?
0
votes
1answer
19 views

What exactly is the difference between Gateaux derivative and directional derivative?

The definition of the limit looks very similar between the two derivatives. It seems that directional derivative is the "amount" of the function going in the direction of a vector (arrow), whereas ...
0
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1answer
21 views

Strong convexity, non-smoothness, and directional derivative

I have a function $f:\mathbb{R}^n\rightarrow \mathbb{R}$ that is (strongly) convex (say in $\mathbb{R}^n$), but not necessarily differentiable. It attains its minimum at $\mathbf{q}$. Given two ...
0
votes
1answer
24 views

Multivariable Chain Rule for partially differentiable maps

The following statement is a simple consequence of the multivariable chain rule: Assume the subsets $X \subset \mathbb{R}^m$ and $Y \subset \mathbb{R}^n$ are open. Consider the maps $$g:X \to ...
4
votes
2answers
81 views

Find a function $f(x)$ in an integral

(Related question here). Is there a way to calculate the function $f(x)$ in this integral in terms of $x$ without using $a,b,c$: $$\int_{a}^{b} f(x)dx=c$$ Two examples $\rightarrow$ how do find ...
0
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0answers
20 views

Do the higher-order Frechet derivatives can be used in optimization?

I am working on inverse problem in seismics. In our community, we use 1-order and 2-order Frechet derivatives a lot to solve the inverse problems being posed. However, some seismologist verified that ...
1
vote
4answers
89 views

Calculate function: $\int_{a}^{b} \left(f{(x)}\right)dx=c$

Is there a way to find the function $f{(x)}$ for a given value of $a,b,c$? $$\int_{a}^{b} \left(f{(x)}\right)dx=c$$ For example: $a=0,b=1,c=\frac{1}{3}$ we get: $$\int_{0}^{1} ...
0
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0answers
22 views

Perturbation theory and variable exchange of poisson-boltmann equation in spherical coordinates

I'm trying to understand this article. I think he has missing terms in his equations, and I can't understand how he derived equations 8-10. The math should be straight forward, and this make ...
4
votes
1answer
46 views

Compute the derivatives of $\frac{d^{2\ell}}{dx^{2\ell}}\tanh(x)^{2k}$ in $x=0$

I would like to compute the derivatives $\frac{d^{2\ell}}{dx^{2\ell}}_{\vert x=0}\tanh(x)^{2k}$ at $x=0$ where $k,\ell\in \mathbb{N}$ positive integers with $\ell\geq k$. I am not sure how to attack ...
0
votes
3answers
42 views

Using Discriminant to find equation of a line?

Find the equation of the tangent to the parabola $$y = x^2 − 5x − 3$$ that is parallel to the line $3x − y − 7 = 0$. I know how to solve this question utilizing differentiation, but I can't think of ...
2
votes
0answers
29 views

High-order total derivatives

I am quite new to total derivatives (up to now I knew the existence of partial derivatives only!). I am dealing some computations that involve second and third-order total derivatives and I have some ...
1
vote
2answers
67 views

Looking for a function that fits a certain criteria

I am looking for a function that fits this description: $$ \frac{d^n}{dx^n}[f(x)] = n! f(x) $$ or $$ \frac{d^n}{dx^n}[f(x)] = (n-1)! f(x) $$ For all values of $n$, with this function i am looking to ...
0
votes
1answer
53 views

Is there a way to visualize, like a picture in mind, the $n$-th derivative?

Is there a way to visualize (like a picture in mind) the $n$-th derivative ? For $n=1$ is the tangent line and we can visualize it quite well. More abstractly is it possible to see the geometric ...
0
votes
1answer
29 views

Am I headed in the right direction with this area optimization question?

Question: A fence will create a rectangular area with one side being formed by an existing building (and hence, the fence only needs 3 sides). One side will be created using Redwood fencing and the ...
2
votes
0answers
25 views

Understand and Implement adjoint operator of gradient

I have an equation as With initial ui=vi; yi=0, tau=1; I am implementing eq. 4.19 and 4.20 in MATLAB. Hence, I would like to ask you something. The adjoint operator of gradient is similar the ...
2
votes
3answers
40 views

limits of integration and derivative

I have an integral that gives $$\left[\frac{d}{dx}[f(x)]\right]_a^b$$ is it possible in general to claim that this is equal to $$\frac{d}{dx}\left(\left[f(x)\right]_a^b\right)\text{ ?}$$ If not in ...
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vote
3answers
57 views

Chain rule for integrals, how?

Can you please give some hints how to solve such a task: Given 3 smooth functions: $f: \mathbb R^2 \rightarrow \mathbb R$, $a,b: \mathbb R \rightarrow \mathbb R$. I should determin the derivative ...
0
votes
1answer
68 views

How to calculate $\int_{0}^e\left(\frac{d}{dx}\sqrt[x]{x}\right)\,{\rm d}x $

I need to evaluate $$\int_{0}^e\left(\frac{d}{dx}\sqrt[x]{x}\right)\,{\rm d}x$$ How would you solve this? What I know is that $\sqrt[x]{x}$ is greatest when $x=e$. That means that $\lim\limits_{x\to ...
0
votes
1answer
36 views

How to find the derivative of $|f(x)|$

The original question was to find domain of derivative of $y=|\arcsin(2x^2−1)|$. First method My attempt was to break $y$ into intervals ,i.e., where $\arcsin(2x^2−1)\geq 0$ and where ...
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votes
1answer
35 views

Differentiation of a modulus function

How to find derivative of $$f(x)=|\sin^{-1}(2x^2-1)|$$ Please provide stepwise mechanism. The original question was to find domain of derivative of y=|arc sin(2x^2−1)|. My METHOD- My attempt was ...
0
votes
1answer
29 views

Differentiating an indirect function

Question: Find $\frac{dy}{dx}$ if: $$x^2 + y^2 = t + \frac{1}{t}$$ and $$x^4 + y^4 = t^2 + \frac{1}{t^2}$$ Attempt: To find $dy \over dx$, we basically need to find $dy \over dt$ and $dx \over ...
2
votes
1answer
62 views

How to solve $x=\sin^{-1}(\frac{1}{2\sqrt{x}})$

I was setting a question when I came across a problem. The question was: Suppose I have a function $y=e^{1+\cos(x)+\sqrt{x}}$. (A) Locate its turning points by taking derivatives and sketch its graph ...
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votes
0answers
26 views

derivative of an equation containing min

I would appriciate if someone could help with letting me know how to take the dervivative of $\large{\frac{dDt}{dg}}$ of the following equation: $${Dt ...