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

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2
votes
0answers
38 views

why do we use dy/dx as ratio though it is not while solving the problems of integration by substitution [duplicate]

According to my knowledge dy/dx is not a ratio. Then while solving the problems of integration by substitution how can we use it as ratio. Because of we have dx/dt =f(x). Then while shoving it by ...
1
vote
0answers
33 views

Symbol of differential operator and change of coordinates

Some time ago I posted the question about the change of coordinates in differential operator. Here is the relevant discussion Symbol of differential operator transforms like a cotangent vector The ...
0
votes
2answers
28 views

Finding the tangent line of a piecewise-defined function

I have $ f(x) = \begin{cases} \frac{e^x-1}{log(x+1)} & \quad \text{if } x>-1 ,&x\not=0 \\ 1 & \quad \text{if } x= 0\\ \end{cases} $ I need the tangent line of ...
2
votes
2answers
71 views

How do I find the derivative of $(1 +1/x)^x $

I tried one approach but the correction in the book shows me a total different answer. Here's what I did: $(1+ 1/x)^x=xln(1+1/x)$ Thus, now we try to find the derivative of a multiplication: $ u(x)=...
0
votes
0answers
8 views

Mean shift with Epanechnikov kernel

The multivariate Epanechnikov kernel is given by $$ K_E(\vec{u}) = c(1-\vec{u}) $$ if $\lVert u \rVert^2 \leq 1$ and $K_E(\vec{u}) = 0$ otherwise. When applying the mean shift algorithm, the update ...
22
votes
2answers
2k views

If $f(x)$ has a vertical asymptote, does $f'(x)$ have one too?

So here is what I understand: If $f(x)$ is increasing/decreasing, then its derivative $f'(x)$ is positive/negative and... If $f(x)$ is increasing/decreasing, then the derivative of $f'(x)$ (...
0
votes
1answer
22 views

Value of $V/(250\pi)$

A cylindrical container is to be made from certain solid material with the following constraints: It has fixed inner volume $V$ mm${}^3$ ,has a $2$ mm thick solid wall and is open at the top. The ...
3
votes
2answers
48 views

Prove that $\overline{f(z)}$ is differentiable at $a \in D(0;1)$ if and only if $f'(a)=0$

Let $f$ be holomorphic in $D(0;1)$ and define $k$ by $k(z)=\overline{f(z)}$. Prove that $k$ is differentiable at $a\in D(0;1)$ if and only if $f'(a)=0$. What I tried was first, assuming $k$ is ...
1
vote
0answers
22 views

Comprehension question about derivative in one point

Find the derivative of $f$ in $(x_{0} , y_{0})^{T}$ for: $$f(x,y)=\binom{x^4+2x^2y^2+y^4}{x^4+2x^2y^2+y^4}$$ Is it right to derivate $\partial x$ and $\partial y$ with $(x_0,y_0)^T$ ...
11
votes
2answers
469 views

Differentiating the binomial coefficient

I took a lecture in combinatorics this semester and the professor did the following step in a proof: He showed that function $f: x \mapsto \binom{x}{r}$ is convex for $x > r - 1$ (in order to use ...
0
votes
3answers
98 views

If $f(x) = x\log2,$ then find $f'(x)?$

I have a function (natural log): $$f(x) = x\log2$$ My textbook shows that the derivative of it is: $$f'(x)=\frac{x}{2}$$ But My teacher told me that we should take the derivative of whatever behind ...
1
vote
2answers
68 views

How to solve $\lim_{x \to 0^+} \frac{x^x - 1}{\ln(x) + x - 1}$ using L'Hôpital

How could I solve $$ \begin{align*} \lim_{x \to 0^+} \frac{x^x - 1}{\ln(x) + x - 1} \end{align*} $$ using L'hôpital? Analysing the limit we have $0^0$ on the numerator (which would require using ...
4
votes
2answers
66 views

Can the second derivative of a function be interpreted as the slope of its “concavity lines”?

Can the second derivative of a function be interpreted as the slope of its "concavity lines"? For example consider the following picture: Does $f''$ for each point $x$ that corresponds to an arrow ...
0
votes
1answer
30 views

Matrix derivative (chain rule application)

Let $x$, $y$ by vectors s.t. $x=f(y)$ and let $B$ be a constant matrix. What is $\frac{\partial x'Bx}{\partial y}$? The partial derivative $\frac{\partial x'Bx}{\partial x}=2Bx$ and we need to use ...
0
votes
0answers
29 views

Discuss the continuity and differentiablity of given function.

If $\big[\cdot\big] $ denotes floor function (i.e the integral part of $x$) and $$f(x)=\big[x \big] \left(\frac{\sin \frac{\pi}{\big[x+3\big]}+\sin \pi \big[x+3\big]}{3+\big[x \big]} \right)$$, then ...
0
votes
1answer
11 views

exercice analysis; inverse theorem, implicit function theorem, locally immersions and submersions, post theorem

Could anyone help me find lists of exercises (in books or other materials) analysis in R for a qualification examination. Threads 0) differentiability in R 1) the inverse function theorem 2) implicit ...
1
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0answers
17 views

About $\frac{\partial^n f}{\partial x^n}$ ,for $f(x)$,What should I think, when $n[\in (\mathbb R\backslash \mathbb Q)^+]$ or $(\in \mathbb C)$? [duplicate]

About $\frac{\partial^n f}{\partial x^n}$ ,for $f(x)$,What should I think, when $n[\in (\mathbb R\backslash \mathbb Q)^+]$ ,pozitive irrational? or $(\in \mathbb C)$ For example; $f(x)=x^2+3x\quad$ ...
-5
votes
2answers
138 views

A new “differential” form for the antiderivative?

The derivative is in general notated by: $\frac {dy}{dx} = \frac d{dx} f(x)$ It has come to my understanding quite recently that dx and dy are actual quantities and not just notational garbage. So ...
2
votes
2answers
96 views

Show that $\text{rank}(Df)(A) = \frac{n(n+1)}{2}$ for all $A$ such that $A^TA = I_n$

We identify $\mathbb R^{n \times n}$ with $\mathbb R^{n^2}$ and define $f:\mathbb R^{n^2} \to \mathbb R^{n^2}, A \mapsto A^TA$. Show that $\text{rank}(Df)(A) = \frac{n(n+1)}{2}$ for all $A$ such ...
3
votes
3answers
90 views

Differentiating $\mbox{tr} (ABA^TC)$ w.r.t. $A$

Why is $\nabla_A \mbox{tr} (ABA^TC) = CAB + C^TAB^T$? Here $A, B, C, D$ are all $n \times n$ matrices. $$\nabla_A f(A) = \left[\begin{matrix} \frac{\partial f}{\partial A_{11}}... \frac{\partial f}{...
0
votes
1answer
18 views

Find the number of root in interval

Let $f,g : \left[-1, 2\right] \rightarrow R $ be continuous function which are twice differentiable on the interval $\left(-1,2\right)$ . Let the values of $f $ and $g$ at the points $-1,0,2$ be as ...
3
votes
1answer
100 views

$f:\mathbb R \to \mathbb R$ be continuously differentiable function such that $f(x),f'(x)>0$ for all real $x$ , then $\lim _{x \to -\infty}f'(x)=0$?

Let $f:\mathbb R \to \mathbb R$ be a continuously differentiable function such that $f(x)>0 , f'(x)>0 , \forall x \in \mathbb R$ , then is it true that $\lim _{x \to -\infty}f'(x)=0$ ? I can ...
-1
votes
1answer
15 views

Help in a rectilinear motion problem in calculus

A particle moves along the $x$-axis according to the equation $$s(t) = \frac 13 t^3 -t^2 -8t +12$$ where $s$ is the directed distance (in meters) of the particle from the origin at time $t$ (in ...
3
votes
1answer
98 views

What is the derivative of $x^{x^{x^{x^{.^{.^{.}}}}}}$ [duplicate]

Here is my attempt: Substituting y for infinite x powers: $$x^{x^{x^{x^{.^{.^{.}}}}}}=y → x^y=y $$ Giving: $$x=y^{\frac{1}{y}}$$ Take natural logs & differentiate with respect to $y$: $$ln(x)=...
1
vote
0answers
16 views

Subscriber left after increase in amount

A telephone company in a town has $ 500$ subscribers on its list and collects fixed charges of $Rs$ $300/-$ per subscriber per year. The company proposes to increase the annual subscription and it is ...
1
vote
1answer
35 views

Consider the function $f(x) = x^2 + 4/x^2$ a) Find$f ^\prime(x)$ b) Find the values of $x$ at which the tangent to the curve is horizontal.

So far I have this... a) $f^\prime(x) = 2x + (0)(x^2)-(4)\dfrac{2x}{(x^2)^2}$ $= 2x - \dfrac{8x}{x^4}$ $= \dfrac{2x^5 - 8x}{x^4}$ $= \dfrac{2(x^4 - 4)}{x^3}$ I believe I derived this correctly. ...
2
votes
5answers
68 views

Differentiate and simplify. $m(x) = \frac{x}{\sqrt{4x-3}}$

My work so far is: \begin{align} m'(x) &= \frac{(1)(\sqrt{4x-3})-(x)(1/2)(4x-3)^{-1/2}(4)}{(\sqrt{4x-3})^2} \\ &= \frac{\sqrt{4x-3} - 2x(4x-3)^{1/2}}{4x-3} \end{align} and now I'm stuck on ...
-2
votes
3answers
59 views

Derivation of a fraction $\frac{f(x)}{g(x)}=\frac{x+8}{9x^2\cdot e^x}$ [closed]

I have been given the fraction of $$\frac{f(x)}{g(x)}=\frac{x+8}{9x^2\cdot e^x}$$ I can derive the $f(x)$ which is equal to $1$, however when I try to derive $g(x)=9x^2\cdot e^x$ I seem to be ...
1
vote
1answer
43 views

How do I find the partial derivatives of heron's formula?

Heron's formula finds the area $A$ of a triangle with sides of length $a$, $b$, and $c$: $$A=\sqrt{s(s-a)(s-b)(s-c)}$$ where $s$ is the semiperimeter of the triangle: $$s=\frac{a+b+c}{2}$$ How do ...
0
votes
0answers
23 views

For $u_h(t) = \frac 1h \int_t^{t+h}u(s)\;\mathrm{d}s$, is $[\partial_t (u_h(t))]_{x_j} = \partial_t[(u_{x_j})_h(t)]?$

Let $u$ belong to $L^2(0,T;H^1(\Omega))$ and $u_t \in L^2(0,T;(H^1(\Omega))^*)$ and define the function $$u_h(t) = \frac 1h \int_t^{t+h}u(s)\;\mathrm{d}s$$ Is it true that $$[\partial_t (u_h(t))]_{...
5
votes
1answer
104 views

Why isn't it mathematically rigorous to treat dx's and dy's as variables? [duplicate]

If I do something like: $$\frac{dy}{dx} = D$$ $$dy = D \times dx$$ People would often say that it is not rigorous to do so. But if we start from the definition of the derivative: $$\lim_{h \to ...
0
votes
1answer
36 views

Simplifying a derivative

I have been trying to calculate the simplified version of the penultimate term to the last one, but I honestly didn't find a way to do that. So, I got to this, but how do I eliminate the ...
4
votes
1answer
110 views

Proving roots of a polynomial are real and distinct.

Let $p(x)$ be a polynomial with all roots real and distinct such none of its roots is equal to zero. Prove that the polynomial $x^2p''(x)+3xp'(x)+p(x)$ also has all roots real and distinct. Unable ...
1
vote
1answer
48 views

Show $f(x) = \sum_{n=1}^\infty \frac{\sin{nx}}{n^{5/2}}$ is $C^1$

The task is to show $$f(x) = \sum_{n=1}^\infty \frac{\sin{nx}}{n^{5/2}}$$ is a continuous function on $\mathbb{R}$ with a continuous derivative. Showing that the series converges at each point is ...
2
votes
2answers
80 views

What is $dy/dy$?

Say you have the following function: $$y=x^2+x$$ Then $$\frac{dy}{dx}=2x+1$$ However, what if you wanted to find $dy/dy$? I differentiated both sides of the original equation with respect to $y$, ...
4
votes
1answer
71 views

Lebesgue integration by substitution

I read that, if $f\in L^1[c,d]$ is a Lebesgue summable function on $[a,b]$ and $g:[a,b]\to[c,d]$ is invertible and such that $g\in C^1[a,b]$ and $g^{-1}\in C^1[a,b]$, then $$\int_\limits{g([a,b])}f(x)\...
1
vote
1answer
41 views

What is the derivative of $Tr(X^{-\frac{1}{2}}D)$ with respect to $X$?

In the question, $X$ and $D$ are symmetric positive definite (SPD) matrices, and $Tr(\cdot)$ is the trace of the matrix. $X^{-\frac{1}{2}}X^{-\frac{1}{2}}=X^{-1}$, and $X^{-\frac{1}{2}}$ is also a ...
16
votes
8answers
3k views

what is sine of a real number

I never understand what the trigonometric function sine is.. We had a table that has values of sine for different angles, we by hearted it and applied to some problems and there ends the matter. Till ...
1
vote
1answer
39 views

Values which makes my function continuos

I have: $$f(x)= \begin{cases} \dfrac{\ln(x+1)-e^x+1}{x}, & x>0 \\ ax, & x \le 0 \end{cases} $$ I need the values who makes the function continuos. I calculated the limit about 0 of the ...
4
votes
0answers
56 views

How to integrate $\frac{x^n\,\,\,\sqrt{x+1} }{\left(m (x+1)^3-x\right)^{3/2}}$?

Let $$ f(x)=\frac{x^n\,\,\,\sqrt{x+1} }{\left(m (x+1)^3-x\right)^{3/2}} $$ $$ 0<m<\frac{4}{27}\,\,\,\,;\,\,\,\frac{1}{2}<n<1 $$ I am trying to integrate the above function. I tried ...
1
vote
5answers
64 views

Why there is multiple root?

My teacher said that If we have$$ f(x)=x^4 $$ Then there will be 4 same root $0$ satisfying the equation . He said that it is because $$f'(x)=4x^3$$ $$f''(x)=12x^2$$ $$f'''(x)=24x^1$$ All are zero ...
1
vote
3answers
153 views

Is there any metric $d$ on $\mathbb R$ and $a \in \mathbb R$ such that the function $f:\mathbb R \to \mathbb R$, $f(x)=d(x,a)$ is differentiable?

Let $d$ be any metric on $\mathbb R$ , then I know that the two variable scalar field $f: \mathbb R^2 \to \mathbb R$ , $g(x,y)=d(x,y)$ is never differentiable . Now what I want to ask is this : Let $a ...
1
vote
2answers
74 views

A function $f$ such that $f' = 0$ when $x < 0$ and $f' = 1$ when $x \geq 0$?

Is there a differentiable function $f: \mathbb{R} \rightarrow \mathbb{R}$ such that $f'(x) = 0$ when $x < 0$ and $f'(x) = 1$ when $x \geq 0$? Attempt: I was trying to think of an example. First ...
0
votes
1answer
35 views

An error in the simplification of a sum

I try to simplify the sum $\sum_{j=0}^{i-2}j3^j$. My method leads to an error. I proceed by evaluating $3(\sum_{j=0}^{i-2}3^j)'$ The derivative $(\sum_{j=0}^{i-2}3^j)'$ is equal to $\left(\frac{3^{i-...
0
votes
1answer
37 views

Why has the equation positive root?

Let $x \in \mathbb{R}$ and $\lambda ,{\lambda _0} \in \mathbb{C}$ and $r\in(0,1)$. $w(x) = {\alpha _m}{x^m} + \cdots + {\alpha _1}{x^1} + {\alpha _0}$. $f(\lambda)$ is function such that $f(\...
0
votes
1answer
29 views

Differentiability complex functions

If u, v : Ω → C are C^1 functions in an open set Ω ⊂ C, and f = u + iv. Then if the Cauchy–Riemann equations is not satisfied at any points of Ω, does it mean that f is not differentiable?
-3
votes
1answer
44 views

Use the definition of derivative for $f(x)=\begin{cases}x^2\sin\frac{1}{x}, \;x\ne 0\\0, \; x=0\end{cases}$

$f(x)=\begin{cases}x^2\sin\frac{1}{x}, \;x\ne 0\\0, \; x=0\end{cases}$ The best I can do so far is getting it into the following form: $\lim \limits_{x \to c}\frac{x^2\sin\left(\frac{1}{x}\right)-c^...
0
votes
2answers
31 views

The greatest value of $\cos(xe^{\lfloor x \rfloor} +7x^2 -3x)$

The greatest value of $g(x)=\cos(xe^{\lfloor x \rfloor} +7x^2 -3x)$ , $x\in [-1,\infty)$ , is My work: For function to be maximum $f(x) = xe^{\lfloor x \rfloor} +7x^2 -3x$ must be minimum When $ x ...
3
votes
2answers
96 views

Show differentiability of $f(x) = \sqrt{x^4 + y^4}$ in $(0,0)$

I'm trying to show the differentiability of $$f:\mathbb R^2 \to \mathbb R\text;\quad f(x) = \sqrt{x^4 + y^4}$$ in (0,0). Here's my attempt: Since $\partial_xf(x,y) = \frac{4x^3}{2\sqrt{x^4+y^4}}$ we ...
0
votes
2answers
37 views

Antiderivative for a function for integration

I have: $$f(x)=\cos(x) \times e^{\sin(x)}$$ and the fitting Antiderivative: $$F(x)=e^{\sin(x)}$$ Can someone please explain to me how I get from $f(x)$ to $F(x)$? In small steps? Thanks!