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
1answer
23 views

Evaluation of derivative: if $p(x)=b_0 + (x-z)q(x)$, then $p'(z)=q(z)$

I just wanted to confirm that I did this correctly, because this answer seemed too easy to obtain: $p(x)=b_0 + (x-z)(q(x)).$ Show that $p'(z)=q(z).$ My answer: $$\begin{align*} p'(x) ...
3
votes
3answers
62 views

If $f$ satisfies $\forall x\in\Bbb{R},0\leq f'(x), f''(x)$ and if $\exists a\in\Bbb{R}$ such that $0<f'(a)$, Then $lim_{x\to\infty}f(x)=\infty$

I got this problem: Let $f:\mathbb{R}\to\mathbb{R}$ be a twice differentiable function that satisfy $\forall x\in\mathbb{R},0\leq f'(x)$ and $0\leq f''(x)$ Prove that if $\exists a\in\mathbb{R}$ ...
3
votes
4answers
73 views

Existence of solution in $x,y \in (a,b)$ of $ \bigg(\dfrac { a+b}2\bigg)^{x+y}=a^xb^y$

Let $a<b$ be positive real numbers , then is it true that there exist $x,y \in (a,b)$ such that $ \bigg(\dfrac { a+b}2\bigg)^{x+y}=a^xb^y$ ?
0
votes
1answer
24 views

Calculating the directional derivative of $ xy^3/(x^3+y^6)$.

$f(x,y)= xy^3/(x^3+y^6)$ if $(x,y)\neq 0$, $f(x,y)=0$ if $(x,y)=0$. Prove that $f'(0; a)$ exists for every vector $a$. I know how to find the directional derivative from limit equation, but don't ...
0
votes
1answer
46 views

Formal explication of $dx/dt = v(x)$ implies that $dt/dx=1/v(x)$

The tittle is all about my question. What is the formal explication of the fact that $dx/dt = v(x)$ implies that $dt/dx=1/v(x)$? Is that via geometry? analysis? differentiable forms? Can you give ...
-1
votes
1answer
33 views

does the following function have all directional derivatives?

$$xy\sin(\frac{1}{xy})$$ the function has partial derivatives at every point , but i wanted to know whether this function had directional derivatives at every point? for $x=0$ the function is ...
1
vote
0answers
47 views

Proving that maximal interval of existence exists and that solution is unque

For each $\lambda\in \mathbb{R}$, let $\varphi_{\lambda}$ : $J_{\lambda}\rightarrow \mathbb{R}$ denote the solution to the following initial value problem: $$ ...
1
vote
2answers
107 views

Finding example of a special type of continuous differentiable function

Give example of a continuous function (if exists) $f : [a,b]\to \mathbb R$ differentiable in $(a,b)$ such that $f(a)f(b) \ne 0$ , the set $A:=${ $x \in (a,b) : f(x)=0$ } is infinite but not an ...
1
vote
1answer
73 views

Second difference $ \to 0$ everywhere $ \implies f $ linear

Exercise 20-27 in Spivak's Calculus, 4th ed., asks us to show that if $f$ is a continuous function on $[a,b]$ that has $$ \lim_{h\to 0}\frac{f(x+h)+f(x-h)-2f(x)}{h^2}=0\,\,\,\text{for all }x, $$ then ...
1
vote
2answers
46 views

Find the derivative of x^1/5 from the definition

I've been trying to figure out how to compute the derivative of $f(x) = x^{1/5}$ at $x=1$ from the definition. Here's what I've done: $$f'(1) = \displaystyle \lim_{\Delta x\rightarrow 0} ...
1
vote
1answer
102 views

There is a unique polynomial interpolating $f$ and its derivatives

I have questions on a similar topic here, here, and here, but this is a different question. It is well-known that a Hermite interpolation polynomial (where we sample the function and its derivatives ...
0
votes
0answers
47 views

How do you find the gradient between two different curves that passes through an arbitrary point between the curves?

Given a graph like this, XC, and ZC is it possible to find YC, and if so, how? fB(x) and fA(x) are some known but different functions at ZB and ZA, respectively. fC(x) is NOT a known function but ...
0
votes
2answers
27 views

How to solve when the unknown is given?

A curve has a gradient function $px^2 - 5x$, where $p$ is a constant . The tangent to the curve at the point $x=1$ is parallel to the straight line $y+2x-5=0$. Find the value of $p$.
1
vote
4answers
83 views

If $f$ is continuous on $[0,\infty)$ and differentiable on $(0,\infty)$ and if $lim_{x\to\infty}f'(x)=0$ Then $f$ uniformly continuous on $[0,\infty)$

I got this problem: Let $f$ be a continuous function on $[0,\infty)$ and differentiable function on $(0,\infty)$ such that $\lim_{x\to\infty}f'(x)=0$. (1) Prove that for each $0<\epsilon$ there ...
0
votes
1answer
72 views

Differential and Integral calculus.

Can anyone here explain me, why do we take the Centre of mass of a conical shell using slant height and $dl$ whereas the centre of mass of a solid cone is calculated using the vertical height and ...
1
vote
1answer
78 views

What does $\ dx^2$ mean?

While writing the second derivative of y, $\frac{d^2y}{dx^2}$ what does the symbol $dx^2$ signify? I know that in case of the first derivative $dy$ means change in y and $dx$ means change in y and ...
2
votes
2answers
73 views

$\displaystyle \frac{d}{dx}2^x$ where $x=0$

I put into Wolfram Alpha: d/dx 2^x Where it told me $f'(x)=2^x\log(2)$. Then I put in d/dx 2^x where x=0 and it said "$\displaystyle \log(2)\approx0.693147$" I know through Wolfram ...
0
votes
1answer
35 views

Find the absolute maximum and absolute minimum values of f on the given interval, f(x) = x^2 e^{-x/2}, [-2,8]

Here's the function: f(x) = x^2 e^{-x/2}, [-2,8] Sorry for asking this question again, but i cant seem to move forward. Can i get some help again? so i graphed the ...
0
votes
4answers
128 views

Finding the derivative of $v(r) = k(R^2 − r^2)$

The velocity (in centimeters per second) of blood r cm from the central axis of an artery is given by $$v(r) = k(R^2 − r^2)$$ where $k$ is a constant and $R$ is the radius of the artery. Suppose $k ...
1
vote
2answers
39 views

Algebraic issues with the calculation of the second derivative of $(a+be^x)/(ae^x+b)$

I'm trying to work out the 2nd derivative of $\dfrac{a+be^x}{ae^x+b}$ I have $f''=\dfrac{(ae^x+b)^2(b^2-a^2)e^x-2ae^x(ae^x+b)(b^2-a^2)e^x}{(ae^x+b)^4}$ There are so many terms, and I'm seriously ...
1
vote
2answers
430 views

How to find the derivative of improper integral with variable upper limit?

I have the integral from $-\infty$ to $y^2$ of the function $(e^{-|x|})$ and I need to find the derivative of this. That is, $$\frac{d}{dy} \int_{-\infty}^{y^2} e^{-|x|}\,dx$$ Usually derivative ...
1
vote
2answers
97 views

Can the derivative of an absolutely continuous real function have a simple discontinuity?

If $f'$ exists everywhere, then we know that it cannot have any simple discontinuities. But in this case we only know that $f'$ exists a.e. (since $f$ is absolutely continuous). More specifically, ...
1
vote
1answer
84 views

Does a word problem provide all information?

A while ago I asked a similar question about word problems and assumptions. Is it a definition or an accepted-fact that word problems provide all information about the relevant existence/situation in ...
2
votes
4answers
60 views

What is the rule behind this derivative?

$$\dfrac{\rm d}{{\rm d}t}\big(\sin^2(t)\big)=\sin(2t).$$ I don't understand what is the rule behind this derivation. I had tried to first rerivate sin() and then to derivate the square function, but ...
8
votes
1answer
236 views

If the set of values , for which a function has positive derivative , is dense then is the function increasing?

Let $f:\mathbb R \to \mathbb R$ be a differentiable function such that $A:=${ $x \in \mathbb R :f'(x)>0$ } is dense in $\mathbb R$ , then is it true that $f$ is an increasing function ? What ...
0
votes
1answer
17 views

On the existence of a non-constant sequence whose differentiable image converges [duplicate]

Let $f: [a,b] \to \mathbb R$ be a function differentiable in $(a,b)$ , then is it true that there is a non-constant sequence $(x_n)$ in $(a,b)$ such that the sequence $\big(f(x_n)\big)$ is ...
0
votes
2answers
28 views

Where am I wrong with this derivative?

I want to derivate this function : $$f(t) = \frac{3}{\sin(t)}$$ I know that the derivative of $\frac{u(x)}{v(x)}$is$\frac{u'v-uv'}{v^{2}}$ in general and that in this fraction : $$u'(t) = 0$$ $$v'(t) ...
0
votes
1answer
163 views

Impatience and interest rate

I'm having difficulties solving the following problem in economics. I come from a mathematical background, and it's hard for me to get some of the terms: Consider a two-period economy with a ...
0
votes
1answer
129 views

Interpretation of differential form

We know what is the interpretation of a total differential, ex.: $$df=\frac{\partial f}{\partial x} dx+ \frac{\partial f}{\partial y} dy$$ But what is the interpretation of a 1-form and its exterior ...
12
votes
3answers
414 views

The closed form of $\lim_{x\to\frac{4}{3}}\frac{\partial}{\partial x}\left[\,_2{\rm{F}}_1\left(\frac{1}{3},1;x;-1\right)\right]$

Do you think the following limit might have a closed form? Some hints or clues? $$\lim_{x\to\frac{4}{3}}\frac{\partial}{\partial x}\left[\,_2{\rm{F}}_1\left(\frac{1}{3},1;x;-1\right)\right]$$
0
votes
0answers
16 views

Differential and a notation problem

Let $df(x) = f'(x)\,dx \:\:\:\:\: (1)$ Now we want to integrate both sides and we get: $f(x) = f(x)$ But now we want to differentiate again and we get $f'(x)=f'(x)$ I just don get it. If we say ...
1
vote
1answer
155 views

Differentiating a Quadratic Form

I'm having some trouble differentiating a quadratic form. I'm tasked with showing that $P(x) = \frac{1}{2} \left(b-Ax\right)^T C (b-Ax)$ is minimized by a vector $x$ satisfying $A^T C A x = A^T C b$. ...
0
votes
1answer
919 views

Continuity and differentiability on piecewise function

Let $$f(x)=\begin{cases}x^2-3, & x<0;\\-3, & x\geq 0.\end{cases}$$ (a) Find the value of $x$ where $f$ is discontinuous (b) Find the value of $x$ where $f$ is non-differentiable ...
3
votes
1answer
89 views

Finding $\int_{0}^{1} \frac{\log(1+x)}{1+x^2} {\rm d}x$ by differentiating under the integral sign.

I've tried to find this integral by the method already outlined in the title. I decided to let $$ \displaystyle I(\alpha) = \int_{0}^{1} \dfrac{\log(1+\alpha x)}{1+x^2} \text{ d}x. $$ From this ...
0
votes
1answer
34 views

What is the derivative of a Radial Basis Interpolation function?

A radial basis interpolation function is described as: $ f(\textbf{x})=\sum_{k=1}^N c_k \phi(\lVert \textbf{x}-\textbf{x}_k \rVert_2), \ \textbf{x}\in\mathbb{R}^s $ where $\textbf{x}_k$ are the $N$ ...
4
votes
5answers
303 views

Simple differentiation from first principles problem

I know this is really basic, but how do I differentiate this equation from first principles to find $\frac{dy}{dx}$: $$ y = \frac{1}{x} $$ I tried this: $$\begin{align} f'(x) = \frac{dy}{dx} & ...
0
votes
0answers
33 views

Integration with matrices

I have written two equations in matrix format as follows $m(t)={\begin{pmatrix} 200\\ 300\\ 400\\ 500 \end{pmatrix}}^T \begin{pmatrix} ...
2
votes
3answers
67 views

If $f$ is continuous on $[a,b)$ and differentiable on $(a,b)$ such that $\lim_{x\to b^{-}}f(x)=\infty$, Then $f'$ is not bounded above in $(a,b)$.

I got this problem: Let $f$ be a continuous function on the interval $[a,b)$ and differentiable on the interval $(a,b)$, Prove that if $\lim_{x\to b^{-}}f(x)=\infty$, Then $f'$ is not bounded above ...
0
votes
0answers
33 views

Negative Gaussian Log Likelihood Minimization - question about scaling and Hessians

I received this question from my niece, and had to admit it's quite over my head. I wonder if anyone here could assist? I am doing a minimisation with the negative of a gaussian log likelihood ...
0
votes
3answers
62 views

Range of function with limit?

I have a function $$f(x)=\frac{2x^2 - x - 1}{x^2 + 3x + 2}$$ from the interval $[0,\infty)$ The limit of this function is $2$. Is the range then simply from $f(0)$ to $2$, and if yes, would I ...
1
vote
0answers
392 views

Differentiation of the Law of Cosines, where a, b, c, A, B, and C are functions of time t

Is the differentiation of the law of cosines ($c^2= a^2 + b^2 - 2ab\cos C$) this? a, b, c, A, B, and C are functions of time t. $$2c \frac{dc}{dt} = 2a \frac{da}{dt} + 2b\frac{db}{dt} - 2b \cos C ...
0
votes
2answers
64 views

Simplification & Differentiation of $\frac{2x}{x^{1/3}}$

Above is the image I had taken a snap shot of. I was working on the problem # 24. I got to rewrite the function as: $y = 2x(x^{-1/3})$ I differentiated it and got the $y'$ as: $y' = ...
1
vote
2answers
37 views

Implicit differentiation with logarithm

Find $y'$ if $y=\ln(7x^2+3y^2)$. I'm kind of confused on this problem and could use everyone's help. Am I supposed to take the derivative first, which is $y=\ln(14x+6y)$? If so, how do I go from ...
0
votes
1answer
38 views

Scalar derivative of quadratic form where matrix depends on variable

I have the expression $$K(p(t),q(t)) = p^T D(q) p$$ Where D(q) is an n x n symmetric matrix, q and p are vectors (n x 1) depending on scalar variable t. I need to take the derivative of K with ...
-1
votes
1answer
45 views

Finding Derivatives of Functions

I've recently been reading a text on classical mechanics and the Mathematics applied to the use of Derivatives and second derivatives, and I was hoping some folks could verify my answers to the ...
1
vote
1answer
48 views

How does this series expand the expression?

How does $$\sqrt{R^2 + |x|^2} = R + \frac{|x|^2}{2R}+\cdots$$ when expanded around the point $x=0$? I tried using a Taylor expansion but it didnt work out.
1
vote
2answers
49 views

Differentiation - simple case

In the book calculus made easy, page 22 the case of the negative power for $y=x^{-2}$ $$\begin{align} y+dy & =(x+dx)^{-2}\tag{1}\\ \\ & = x^{-2}\left(1+\frac{dx}{x}\right)^{-2}\tag{2} ...
6
votes
1answer
156 views

What is the mathematical truth behind the Leibniz notation in differentiating twice or more?

So $f: \mathbb{R} \to \mathbb{R}$ is $n>1$ (or more) times differentiable. The notation of the first derivative makes perfect "sense" with regard to what's going on: $$\lim_{h \to 0} ...
1
vote
2answers
48 views

Derivative at 4, when $f(x)=\frac{1}{\sqrt{2x+1}}$

Derivative at 4, when $f(x)=\frac{1}{\sqrt{2x+1}}$ I choose to use the formula $\displaystyle f'(x)=\lim_{x\rightarrow a}\frac{f(x)-f(a)}{x-a}$ Which after some work I found to be ...
2
votes
2answers
55 views

Partial derivative in two dimensions

I am struggling with section 3.3 of the following thesis https://smartech.gatech.edu/xmlui/bitstream/handle/1853/29610/grigo_alexander_200908_phd.pdf. Page 21 is fine, then the problems occur in ...