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

learn more… | top users | synonyms (2)

1
vote
2answers
33 views

Solution verification for $y'$ when $y=\sin^{-1}(\frac{2x}{1+x^2})$

I was required to find $y'$ when $y=\sin^{-1}(\frac{2x}{1+x^2})$ This is my solution. Above when I put $\sqrt{x^4-2x+1}=\sqrt{(1-x^2)^2}$ then I get the correct answer but when I put ...
3
votes
0answers
51 views

Problem about $\lim \limits_{x \to c} f'(x) = l $ implies $f'(c) = l$

I found this problem in a paper. Let $f$ be a function differentiable on $(a, b)$ except possibly at $c \in (a, b)$. Suppose that $\lim \limits_{x \to c} f'(x) = l \in \Bbb R$. Prove that $f$ is ...
0
votes
2answers
56 views

Rudin's Chain Rule

Rudin's chain rule theorem goes like this: Suppose $f$ is continuous on ${[a,b]}$, $f'(x)$ exists at some point $x\in [a,b], g$ is defined on an interval $I$ which contains the range of $f$, and ...
2
votes
2answers
51 views

Longest pipe that fits around a corner. [duplicate]

While studying, I came upon the problem "Two corridors of widths $a$ and $b$ intersect at right angle. What is the length of the longest pipe that can be carried across the two corridors, touching the ...
1
vote
0answers
60 views

When can I calculate a derivative in a point?

Okay the title makes no sense. I have a two variable function, $f(x,t)$. When is it that $$ \left(\frac \partial{\partial x} f(x,t) \right)\bigg| _{t=0} = \frac{d}{dx} f(x,0)$$? My guess is that it ...
5
votes
2answers
38 views

Small derivative and the measure of a set.

Suppose that $f:\mathbb{R}\to\mathbb{R}$ is a differentiable function, and that on some interval $(a,b)$, $|f'|\leq1$. Is it true that for all measurable sets $E\subset(a,b)$, ...
1
vote
1answer
35 views

Difference between a Fréchet derivative and a total derivative

I've heard many times that they are somehow similar and in some cases mean the same thing. Consider this function: $$f(x,y)=x^2y$$ I have to calculate the Fréchet derivative $f'(x_0,y_0)$ and some ...
1
vote
2answers
34 views

Ratio of CDF to PDF increasing?

Let $\Phi(x)$ be a cumulative normal distribution function and $\phi(x)$ the associated probability density function. Is the ratio $\frac{\Phi(x)}{\phi(x)}$ increasing in x? Numerically it seems to ...
2
votes
2answers
36 views

When it comes to using derivatives to graph, do I have all of these steps right?

Perhaps this is a silly question, but I haven't been able to find a clear answer anywhere as to what exactly the steps are for using derivatives to find the shape of a graph (I'm having difficulty ...
-2
votes
2answers
49 views

Roots of infinitely differentiable function [closed]

let $f:\Bbb R \to \Bbb R$ be an infinitely differentiable function that vanishes at $10$ distinct points in $\Bbb R$. Suppose $f^{(n)}$ denotes the $n$-th derivative of $f$ , for $n\ge1$. Which of ...
1
vote
1answer
33 views

Angle of intersection of the given curves.

What is the angle of intersection of $$[|\sin x| + |\cos x|]$$ And the curve $$ x^2 + y^2 = 5 $$ where $[n]$ denotes greatest integer function. This is a homework question. I have tried to find the ...
-2
votes
0answers
63 views

Differentiation by applying the Leibniz rule

I am having trouble differentiating an equation. I know you need to use Leibniz rule for it, however, I am not sure how to implement it: $$w^*=bx+\int_{w*}^{w_\max} (w-w^*)dG(w)$$ I need to ...
6
votes
3answers
93 views

Derivative Of $\ln(x)$

It is required to find the derivative of the natural logarithm of $x$: $\frac {d}{dx}\ln(x)$ My solution: Let $f(x)=\ln(x) $ then $f'(x)=\frac {d}{dx}\ln(x) $ By definition:$$f'(x)= \lim_{h\to ...
-2
votes
2answers
117 views

How is this derivative paradox solved? [closed]

We have $$x^2=x+x+x+\cdots+x$$ with $x$ terms in the sum and $x\in\mathbb{Z}$. Taking the derivative of the above equation: $$2x=1+1+1+\cdots+1$$ again with $x$ terms. This implies $$2x=x$$ How ...
0
votes
1answer
15 views

How to find singular points of a function without knowing the graph?

Problem: Let $f(x) = (x-1)^{2/3} - (x+1)^{2/3}$. Locate and classify all local extreme values of this function. Determine whether any of these extreme values are absolute. Attempt at solution: We ...
0
votes
0answers
18 views

derivative of indicator function composed with a relaxation of a heaviside function

I want to compute the following: $|\partial_t I_{\{H(u(x),t) \geq \mu\}}|,$ where $I$ represents the indicator function and $H(u(x),t)$ is actually a smoothly relaxation of another indicator ...
3
votes
2answers
155 views

Tangent to the curve

What is the equation of the tangent to the curve $$y = x^{1/3}$$ at the point $(0,0)$ ? This is a homework question. I tried solving it. The derivative comes out to be infinite at the given point. ...
2
votes
0answers
33 views

How to find derivative of $\left\Vert x-a\right\Vert ^{p}:\mathbb{R}^{n}\rightarrow\mathbb{R}$?

Here is what I've tried: $\alpha\left(x\right)=\sum_{i=1}^{n}\left(x_{i}-a_{i}\right)^{2},\ \ \beta\left(y\right)=\sqrt{y}, \ \ \ \gamma\left(z\right)=z^{p}$. Clearly: $\left\Vert x-a\right\Vert ...
0
votes
1answer
24 views

Question regarding differentiability of function at x=0

I removed the absolute signs ,but in D part how do i remove the outer absolute value symbol
1
vote
2answers
63 views

Regarding differentiability at $x =0$ [closed]

Let $f : \Bbb R \to \Bbb R$ be defined by $f(x) = \begin{cases} x^2 \text{ if $x$ is rational}\\ x^4 \text{ if $x$ is irrational} \end{cases}$ Is $f$ differentiable at $x = 0$? How do I begin ?
3
votes
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
votes
0answers
27 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
vote
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
45 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
votes
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
55 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
34 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 ...
1
vote
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
45 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 ...
1
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
votes
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 ...