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

learn more… | top users | synonyms (2)

7
votes
3answers
194 views

Derivative of the magnitude of a vector. Does it exist, or not?

I have a puzzling situation involving derivatives. I want to derivate: $$ \frac{d}{dx}| \mathbf F(x)| $$ This was actually something involving physics. Lets be 2-dimensional for simplicity. Let a ...
0
votes
1answer
57 views

Can this $dx$ be taken out?

I have this expression: $$\frac{1}{f(x)}\frac{df(x)}{dx}=\frac{dg(x)}{dx}$$ Can the two $dx$ be "simplified"? Namely, to get $$\frac{df(x)}{f(x)}=dg(x)$$ Is this right? If it is, what is the ...
0
votes
2answers
45 views

Find $\frac{dy}{dx}$ if $x^3 + x^2y + xy^2 + y^3 = 81$

I need to find $\frac{dy}{dx}$ if $x^3 + x^2y + xy^2 + y^3 = 81$ I am trying to first get y in terms of x, but that is quite lengthy and feels like I am doing something wrong. How do I go about this ...
4
votes
0answers
56 views

More convenient form of derivative of $\mathrm{sinc}(x)$

$\mathrm{sinc}(x)$ is defined as $\frac{\sin(x)}{x}$ except continuous at $x=0$ (insert the removable singularity). The derivative of $\mathrm{sinc}(x)$ is usually given as the derivative of $\frac{\...
0
votes
0answers
32 views

Understanding a calculation deduced for the function $\pi^{-s/2}\Gamma(s/2)\zeta(s)$

With my current knowledges I don't know if this is a bad question, but since I am interesting in this kind of calculations I want to ask you, if I was wrong or if if my statement is obvious. From ...
1
vote
1answer
25 views

How to calculate $∇(r^2/(2z(1+a/z^2)))$ in cylindrical coordinates

How to calculate $$∇\bigg(\frac{(ρ^2)}{2z(1+\frac{a}{z^2})}\bigg)$$ where the function is in cylindrical coordinates $$ρ^2=x^2+z^2$$ $$∇\bigg(\frac{x^2+z^2}{2z(1+\frac{a}{z^2})}\bigg)$$ Is the ...
1
vote
2answers
20 views

Sign of the derivative $ -e^{\frac{1}{2x+2}}\left(sgn\left(x\right)+\frac{1-\left|x\right|}{2\left(x+1\right)^2}\right) $

Good morning to everyone. I have a problem with finding the sign of a derivative: $$ \frac{d}{dx}f(x)=-e^{\frac{1}{2x+2}}\left(sgn\left(x\right)+\frac{1-\left|x\right|}{2\left(x+1\right)^2}\right) $$ ...
0
votes
6answers
40 views

Critical points of a cubic function

There is a function $x^3 - 6x^2 + 9x + 1$. Its critical points are $1$ and $3$. I am very confused, if these points are maximum and minimum points respectively or are both inflection points. Can ...
0
votes
1answer
18 views

Derivative of dot product with transposed function

According to this post Derivative of dot product I have a similar task: $$\langle f(x),g(x) \rangle = f(x)g(x)^T=j(x)$$ I have to show: $j'(x)=g'(x)f(x)^T+g(x)^Tf'(x)$ I know how to ...
1
vote
0answers
11 views

Mean continuity of gradient

Let $f:\mathbb R^n\longrightarrow R$ be a differentiable function, and suppose $\nabla f$ is bounded. Prove that $$\lim_{r\to 0}\frac{1}{\omega_n r^n}\int_{B_r(x)}[\nabla f(y)-\nabla f(x)] dy=0.$$ ...
-1
votes
1answer
24 views

How can I prove that $(0,0)$ saddle-type inflection point of $x|y|+y|x|$?

How can I prove that (0,0) is saddle-type inflection point? the function is: $f(x,y)=x|y|+y|x|$ How can I find the second derivative by $x$ (for the hessian matrix)
0
votes
1answer
80 views

Why can this differential equation be written in $3$ different ways?

Suppose we have the following differential equation using operator notation: $$(D-x)(D+x)y=0\tag{1}$$ where $$D=\frac{d}{dx}$$ Now I could rewrite $(1)$ as $$\begin{align}\require{enclose}(D-x)(D+x)y&...
1
vote
2answers
109 views

Examples of (not) uniformly continuous, non-differentiable, non-periodic functions

Let $I\subseteq\mathbb{R}$ and $f:I\to\mathbb{R}.$ $(0)$ If $f$ is discontinuous on $I$, then it is not uniformly continuous. $(1)$ Suppose $I$ is open and bounded. If $f$ is unbounded on $I$,...
3
votes
1answer
51 views

function is not differentiable on $\mathbb R\setminus\{0\}$

I need to prove that the given function $f$ is not differentiable on $\mathbb R \setminus\{0\}$. $$ f(x) = \begin{cases} x^2, \ x \in \mathbb{Q}\\ 0, \ x \in \mathbb{R}-\mathbb{Q} \end{cases} $$ ...
0
votes
0answers
36 views

Derivatives that are tangent to the original function

I was recently studying parabolas $ f(x) = ax^2 + bx + c $ whose derivative $f'(x) = 2ax + b$ is tangent to itself -- one example would be $f(x) = x^2 -6x +10;$ it is easy to see that if $c = a + \...
0
votes
2answers
41 views

Intuition behind the derivative of are of a square? How to properly use the derivative ?

If I derive the formula $$S=16t^2$$, where S denotes the distance and t denotes time I get $$ds/dt= 32t$$. This in return give me a formula for the speed of the object at any time t. However if we ...
1
vote
1answer
43 views

Derivative of $\arcsin \frac{x-1}{x+1}$

I was looking at a question that asks for the derivative of $\arcsin (\frac {x+1}{x-1}) $. The solution starts by saying $y = \frac{x+1}{x-1}$, so $1-y^2= \frac{4x}{(x+1)^2}$ and $\frac{1}{\sqrt{1-y^...
0
votes
1answer
47 views

How many solutions does $a^x=2016x$ for $a > 0$ have?

How many solutions does $a^x=2016x$ for $a > 0$ and $x \in \mathbb{R}$ have? Note that for $x < 0$ we have $a^x > 0$ and $2016x < 0$ so we can consider only $x \ge 0$. Let $f(x) = 2016x - ...
0
votes
0answers
12 views

Apply Laplace Operator with two functions

Proof that $\Delta u(x,y):=\partial_1\partial_1u(x,y)+\partial_2\partial_2u(x,y)=0$ is valid for $\log(x^2+y^2)$ and $\arctan\left(\frac{x}{y}\right)$. Is it enough to differentiate the functions in ...
1
vote
1answer
25 views

Derivative of dot product of Residual Sum Square in matrix notation

I am trying to derive the following expression w.r.t. $\beta$: \begin{equation} RSS(\beta) = (\mathbf{y} - \mathbf{X} \beta)^T (\mathbf{y} - \mathbf{X} \beta) \end{equation} I know that the ...
0
votes
1answer
49 views

Analytical solution for a Non-linear differential equation $\frac{d^2y}{dt^2} = A\left(\frac{dy}{dt}\right)+B \sin(2Cy)$

Analytical solution for a non-linear differential equation: $\frac{d^2y}{dt^2} = A \left(\frac{dy}{dt}\right)+ B \sin(2Cy)$ A,B are non-zero constants and y (position) is a scalar-value parameter ...
0
votes
1answer
20 views

computing the directional derivative by definition

The definition of directional derivative: \begin{equation} \frac{\partial f(\mathbf{x})}{\partial \mathbf{v}} = \lim_{h \rightarrow 0} \frac{f(\mathbf{x} + h \mathbf{v})-f(\mathbf{x})}{h}. \end{...
0
votes
2answers
49 views

Integrating product of Dirac delta function and it's derivative

Is the following true: $$ \int \delta(x-a) \frac{\partial^l \delta(x)}{\partial x^l} dx = \frac{\partial^l \delta(x)}{\partial x^l} \Bigg |_{x=a}$$ If not, is there a correct way to evaluate the left ...
0
votes
2answers
44 views

Antidifferentiation: Stone dropped from $150ft$ rising at $10ft/sec$

A stone is dropped from a balloon when it is $150ft$ above the ground and rising at the rate of $10ft/sec$. How long will it take the stone to strike the ground, and with what velocity does it strike ...
2
votes
4answers
51 views

Antiderrivative of ${d^2 y \over dx^2} = 1-x^2$

At any point $(x,y)$ on a curve, ${d^2 y \over dx^2} = 1-x^2$, and an equation of the tangent line to the curve at the point $(1,1)$ is $y=2-x$. Find an equation of the curve. This is what I've done ...
1
vote
3answers
43 views

Differentiability of piecewise functions

Check whether the function is differentiable: $$f:\mathbb{R}^2\rightarrow \mathbb{R}$$ $$f= \begin{cases} \frac{x^3-y^3}{x^2+y^2} & (x,y)\neq (0,0) \\ 0 & (x,y) = (0,0) \\ \end{...
0
votes
0answers
43 views

Calculus: Derivative of a summation and dot product

I'm trying to implement a speed boost to an eye-tracking algorithm (found here: http://www.inb.uni-luebeck.de/publikationen/pdfs/TiBa11b.pdf). I need to take the derivative of the eye-tracking ...
2
votes
1answer
51 views

Neural Network - Why use Derivative

Good Day I am trying to get an understanding of Neural Network. Have gone through few web sites. Came to know the following: 1) One of main objective of neural network is to “predict” based on data....
0
votes
1answer
44 views

Differential equation without analytic solution - comparative statics

I am facing a differential equation - with boundary condition $v(T)$ given - without an analytic solution but still need to understand how the solution is affected by a change of the function's value. ...
1
vote
1answer
49 views

Concavity of function $\sin(x^2)$.

I want to check where the function is convex and where concave. For this I need to calculate the second derivative test: I got $f''(x) = 2*\cos(x^2) -4*x^2(\sin(x^2))$ and this derivative should be ...
0
votes
1answer
24 views

Chain rule confusion(multivariable functions)

I am confused bz the chain rule of multivariable function. I know, that sometimes it is impossible to dot it. For example i have the following functions: $f(x_1,x_2) = \begin{bmatrix} x_1x_2^2 + x_1^...
2
votes
1answer
100 views

Why is $\frac{d(x^n)}{d(x)}=nx^{n-1}$

So I was thinking about what I have learnt and I realised that I kind of took the derivative of a function for granted. So I did some research as I wanted to find out how this was discovered and I ...
0
votes
2answers
33 views

Confusion about the different ways of writing Taylor Polynomials

For the sake of using a simple example, let's say I want to approximate $y=x^3$ with a second degree polynomial, and let's say I want to construct my polynomial around the point $x=4$. One way I ...
0
votes
3answers
30 views

Derivatives: Combining Product & Chain Rules

So I'm working through the material on Khan Academy, and the question is: "Consider the function $f(x) = x^n\ln x$, defined for $x > 0$. Determine, in terms of $n$, the value of $x$ for which $f'...
1
vote
2answers
49 views

Question about proof: continuity of partial derivatives implies total differentiability

I have a lack of understanding regarding this proof, and since the proof is not in English, I will simply write it down up to the point where I can't go further: Statement: Assume $U \subset \Bbb R^...
1
vote
1answer
30 views

Bounding a $C^0$ function with $C^1$ functions

Given a continuous function from $(0,+\infty)$ in itself, with $\lim_{x\to 0^+} f(x)=0$, find $C^1$ functions $g,h:(0,+\infty)\longrightarrow(0,+\infty)$ such that $g\leq f\leq h$ and $\lim_{x\to 0^+}...
1
vote
2answers
42 views

Partial Derivatives Approximation

By definition we know the following: \begin{equation} \frac{\partial f(x,y)}{\partial x} \approx \frac {f(x+ \delta x,y)-f(x,y)}{\delta x} \end{equation} \begin{equation} \frac{\partial f(x,y)}{\...
-1
votes
0answers
14 views

Is there any equality for the integral of the product of normal derivative?

I am trying to get the proof of $\int\int_DD_uf(x) D_ug(x) dx$. For example in Green Theorem, in integral we use the product of $ \nabla$, when it comes to normal derivative, how can I organize the ...
2
votes
1answer
90 views

Second Differential

Let $(x,y,z)$ a coordinate system, $M=\mathbb{R}^3$ and we also denote by $x$ the first coordinate function : $x:M \rightarrow \mathbb{R},\; q=(a,b,c) \mapsto a$. We have $dx:TM \rightarrow \mathbb{R}...
0
votes
2answers
95 views

Minimum value of $4a+b$

Let $ax^2+bx+8=0$ be an equation which has no distinct real roots then what is the least value of $4a+b$ where $a,b\in \Bbb R$. My Try: I differentiated the given function to get $f'(x)=2ax+b$ now ...
-1
votes
0answers
57 views

Analytical solution for a Non-linear differential equation $\frac{d^2y}{dt^2} = A\left(\frac{dy}{dt}\right)+[B \sin(Cy)\times\cos(Dt)]-E \sin(2Cy)$

Is there any analytical solution for the following differential equation? $\frac{d^2y}{dt^2} = A\left(\frac{dy}{dt}\right)+[B \sin(Cy)\times\cos(Dt)]-E \sin(2Cy)$ A,B,C,D are non-zero constants and ...
0
votes
0answers
11 views

Differentiation of multivariable functions(inverse and chain rule) [duplicate]

In my case i have the following functions: $f(x_1,x_2) = \begin{bmatrix} x_1*x_2^2 + x_1^3*x_2\\x_1^2*x_2 + x_1 + x_2^3\\\end{bmatrix}$ $g(u) = \begin{bmatrix} e^u \\ u^2 + u\\\end{bmatrix}$ An i ...
6
votes
4answers
835 views

Why doesn't derivative difference quotient violate the epsilon-delta definition of a limit?

So the difference quotient is defined as: $$\lim \limits_{h \to 0} \frac{f(x+h)-f(x)}{h}$$ So if we take a function such as $f(x)=x^2$ and go through the simplification, we get $$\lim \limits_{h \...
0
votes
1answer
39 views

Finding Partial Derivative in two ways

I am supposed to find $f_x(0,0)$ of $\frac{5x^2y}{x^4+y^2}$, EDIT: which has a defined value of $0$ at $(0,0)$. The way I did it, I first found the general expression for $f_x(x,y)$, which is $$f_x(...
2
votes
1answer
33 views

Find Polynomial of order 10 for $f(x)=sin(x)$ near x=0

My work so far : I presume the answer should look more like a summation? Thanks!
1
vote
5answers
101 views

With the linear approx. of $f(x)= sin(x)$ around $0$ Calculate $\lim_{\theta\to 0} \frac{sin\theta}{\theta}$

With the linear approximation of $f(x)= sin(x)$ around $0$, calculate: $$ \lim_{\theta\to 0} \frac{\sin\theta}{\theta}$$ Figured I have to use L'Hospital's Rule, but I think I don't get how to ...
0
votes
0answers
26 views

Which functions can be meaningful differentiable?

I have two functions: $f(x_1,x_2) = \begin{bmatrix} x_1*x_2^2 + x_1^3*x_2\\x_1^2*x_2 + x_1 + x_2^3\\\end{bmatrix}$ $g(u) = \begin{bmatrix} e^u \\ u^2 + u\\\end{bmatrix}$ The questions is:which of ...
1
vote
1answer
24 views

Find intersection angle of curves : $y=x^3-5, y=5x^2-x$

My work so far looks as above. Calculated angle seems too large? Not sure what's wrong in here with my calculations. Thanks!
-1
votes
2answers
45 views

Real analysis: Continuity and Differentiability [closed]

Let $f(x)=x^2$ if $x$ is rational and $f(x)=0$ if $x$ is irrational. a) Prove that f is continuous at exactly one point, namely $x=0$. b) Prove that f is differentiable at exactly one point, namely $...
3
votes
1answer
69 views

Statements about derivatives and integrals [closed]

My professor gave me one example. It's given one intervall $I=\left [ a,b \right ]\subset \mathbb{R}$ and one function $f:I\mapsto \mathbb{R}$. There is also given 8 statements about derivatives ...