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

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If $|a+b+p+q|=\frac{k}{18}$, then find the value of $k$

Let $$f(x)= \begin{cases} ax(x-1)+b & x<1 \\ x+2 & 1\leq x\leq 3 \\ px^2+qx+2 & x>3 \end{cases} $$ be continuous for all x except $x=1$ but $|f(x)|$ is ...
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4answers
52 views

Derivative of degree k for $f(t)$ $=$ $1 \over {1 + t}$

Given $f: \Bbb R \setminus \{-1\} \rightarrow \Bbb R$, $f(t)$ $=$ $1 \over {1 + t}$, I would like to calculate the derivative of degree $k$. Approach First, we try to examine if ...
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0answers
55 views

Prove $\Bigg(\langle\nabla f(x),x \rangle = af(x) \Bigg) \Leftrightarrow \Bigg(f(tx)=t^af(x) \Bigg)$ for $f: \Bbb R^n \to \Bbb R$ differentiable

Let $n\in \Bbb N , a \in \Bbb R$ and $f: \Bbb R^n \to \Bbb R$ differentiable. I have to show: $$\Bigg(\langle\nabla f(x),x \rangle = af(x) \:\:\:\forall x \in \Bbb R ^n \backslash\{ 0 \} \Bigg) \...
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1answer
29 views

If the volume of a cylinder is fixed, derive the radius and height that will maximize the surface area

I know how to find the radius and height for minimum surface area. [https://www.physicsforums.com/threads/maximum-surface-area-of-cylinder.332279/ ]. For it to be minimum, h=r/2. It would be great if ...
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0answers
45 views

Concavity when 2nd derivative is zero

I was self-studying for a CLEP calculus exam when the following problem came up. It basically asks you to sketch a graph of a function based on the information given. My question is about the part ...
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2answers
51 views

Two variable definition of derivative

Let $f:(0,1)\rightarrow \mathbb R$ be a real valued map from the unit interval. Let $$A:=\left \{a\in (0,1):\exists f^*(a)=\lim_{x\ne y,\,(x,y)\to(a,a)} \frac{f(x)-f(y)}{x-y}\right \}$$ It is ...
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1answer
73 views

Why does the gradient of matrix product $AB$ w.r.t. $A$ equal $B^T$?

The below passage is from p. 215 of Deep Learning by Goodfellow, Bengio and Courville. For example, we might use a matrix multiplication operation to create a variable $C = AB$. Suppose that the ...
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3answers
69 views

what does $(A\cdot\nabla)B$ mean?

I was studying a physics book and I saw this expression $$(A\cdot\nabla)B$$ where $A$ and $B$ are vectors. What's the definition of this? I've also seen this in some identities
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5answers
114 views

Why every point of a function where differentiation exists has only one tangent?

Can anyone help me out? Why every point of a function where differentiation exists has only one tangent? I know the slope at any point of any function is defined by differentiation at that point.But ...
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1answer
50 views

Continuous function with non-negative derivative a.e. implies non-decreasing?

Let $f \colon [a,b] \rightarrow \mathbb{R}$ be a continuous function on a compact interval of the real line. Suppose that $f$ is differentiable almost everywhere and that $f'(x) \geq 0$ at every point ...
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1answer
45 views

If $f(x) = x^3 + 4x^2 + ax + 1$ is a monotonically decreasing function of $x$ in $(-2, -\frac{2}{3})$ then find $a$

If $f(x) = x^3 + 4x^2 + ax + 1$ is a monotonically decreasing function of $x$ in the largest possible interval$ (-2, -\frac{2}{3})$ then find $a$. My work: $$f'(x)=3x^2+8x+a$$ For it to be ...
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1answer
41 views

An alternative formula for a second order Taylor expansion?

I read in a book that the second order Taylor expansion of a function (around $x^0$) can be written as: $$f(x)=f(x^0)+\sum_{j=1}^n df(x^0)/dx_j*(x_j-x_j^0)+\sum_{j=1}^n\sum_{i=1}^nd^2f(x^1)/dx_idx_j*(...
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1answer
26 views

2nd derivative of xy w/ respect to x?

$$\frac{d^2}{dx^2}xy$$ I know it equals zero but I don't know the in between-steps. I'm using it to prove Newtons Laws work in any frame of reference. So say two guys start from the same point and ...
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1answer
32 views

Find number of tangent to given curve

The number of tangent to curve $x^\frac{3}{2} +y^\frac{3}{2} = a^\frac{3}{2}$ where the tangents are equally inclined to axes, is My work $$\frac{dy}{dx}=-\sqrt\frac{x}{y}$$ From above we can say ...
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1answer
22 views

Could this be proper notation for an antiderivative? Does this notation even exist?

If we define $f(x)$ as some arbitrary function, then we can define $f'(x)$ or $f^{(1)}(x)$ as the first order and $f''(x)$ or $f^{(2)}(x)$ as the second order. My question is: Is there sure thing ...
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1answer
14 views

How is the class related to derivability?

Good evening to everyone. I have a question where they require me to find the derivability. After I read the answer sheet I saw that the function has the class $ C^1 $. How is the class related to ...
3
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1answer
28 views

Why $\frac{d}{dt}f(x+t(y−x))<0$ if $x < y, f(y) < f(x)$

Here excerpt from a book: Аssume that $f$ satisfies $\nabla f(x) \ge 0$ for all $x$, but is not nondecreasing, i.e., there exist $x,y$ with $x < y$ and $f(y) < f(x)$. By ...
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2answers
92 views

$f: \Bbb R^2 \to \Bbb R$ whose partials exist. Show: $\partial _xf \:\:\mathrm{continuous} \Rightarrow f \:\:\mathrm {differentiable}$

Let $f: \Bbb R^2 \to \Bbb R$ be a function whose partial derivatives exist. Now i have to show: $$\partial _xf \:\:\mathrm{continuous} \Rightarrow f \:\:\mathrm {differentiable}$$ Any tipps on how ...
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2answers
51 views

How to prove differetiability in $\Bbb K^2$?

I have to investigate differentiability in all points of the following function: $$f: \Bbb {R}^2 \to \Bbb R \: \: \: \: \: \: \: f(x,y):=\begin{cases} y-x &\mbox{if } y\ge x^2 \\ 0 & \mbox{if }...
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1answer
104 views

Proof from Calculus 1

Last days, from going into a website of the university of Pisa, I found an exercise given in the previous exams, in 1999. The problem was like: Given a continuous function $f$ in $\mathbb R$, and ...
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1answer
38 views

How this integral is evaluated $\frac{\partial }{\partial x}\left(\int _y^x\cos \left(-5t^2-2t-4\right)\:dt\right)$?

How this integral is evaluated? $$\frac{\partial }{\partial y}\left(\int _y^x\cos \left(-5t^2-2t-4\right)\:dt\right)$$ And in general, are there general methods for partial differentiation ...
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2answers
56 views

Where have I gone wrong in finding the derivative of $2^{\sin x}$?

I was finding the derivative of $2^{\sin x}$. My attempt $(1)$- $$y=2^{\sin x}$$ $$\implies\ln y=\ln2^{\sin x}$$ $$\implies\log_ey=\ln2^{\sin x}$$ $$\displaystyle\implies e^{\ln2^{\sin x}}=y$$ $$...
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1answer
37 views

Using the chain rule for cos and sin functions

I am having issues with derivatives containing chain rules. I know there is multiple threads already but after reading a few, I still find myself confused. I also checked the actual answer following a ...
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1answer
51 views

Definition of derivative to calculate $x\sqrt{|x|}$ at $x=0$

So the question says use the definition of the derivative to calculate the derivative of $x\sqrt{|x|}$ at $x=0$. I understand the definition of derivative but have no idea where to go from there to ...
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2answers
31 views

Why is it true that $S'(t)/S(t) = d log(S(t)) / dt$?

I came across this identity in derivation of the hazard rate in survival analysis.
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1answer
27 views

Derivative of a characteristic polynomial at an eigenvalue

Let $p(\lambda)$ be the characteristic polynomial of an $n\times n$ matrix $A$. We know that the roots of $p(\lambda)$ are the eigenvalues of $A$, hence the sum of the roots of the polynomial (taking ...
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How would I find the nth derivative of a function, where n is imaginary? What about where n is not a constant? [duplicate]

Forgive me if this question has already been asked. I was unable to find anything relevant to this question. The $n$th derivative of a function, $f^n(x)$ is well-defined for $n\in\mathbb{Z}^+$. As ...
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1answer
30 views

What should i conclude from the following workout?

We know that for any value of $x$ other than $0$, $a^x\ne e^x$ where $a>e, a\in R^+$ but we do know that for some value of $p$, $$pa^x=e^x\ldots(1)$$ you see $p$ is a positive number because of ...
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0answers
34 views

Exercise at Differential Equations: derivatives of boundary conditions

$ Let\quad U\quad be \quad a \quad smooth \quad solution \quad of \quad the \quad following\quad boundary \quad value\quad problem: $ $ -cU'+ (F(U))'=εU''\qquad U(-\infty)= A \quad and \quad U(+\...
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0answers
26 views

Derivative of projection

Let $\Omega \subset \mathbb{R}^n$ be a limited domain of class $C^\infty$ (open, connected) and $\varepsilon > 0$ small such that $$ \Omega_\varepsilon = \{y \in \overline{\Omega} : d(y, \partial \...
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0answers
17 views

Polynom subspace of continuously differentiable Functions

Let $n\in \mathbb{N}$ and $a\in \mathbb{R}$. Then $\mathcal{C}^n(\mathbb{R})=:V$ and $$\langle f,g\rangle :=\sum_{k=0}^n {f^{(k)}(a)g^{(k)}(a)}$$ is a positive semidefinite Bilinear Form for all $f,g\...
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1answer
76 views

Show relation and linearity related to differentiable functions

I have problems solving the following exercise: (a) Let $n\in \mathbb N$, $a\in \mathbb R$ and $f:\mathbb R^n \backslash \{ 0 \} \to \mathbb R$ $\mathbb R$-differentiable. Show that the relation $$...
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1answer
20 views

Show Each function is equivalent using 2 conditions.(Real analysis)

It is might be easy for you. The Question There are functions $ f, g, c, s $ $f,g : R \rightarrow R $ and $s,c : R \rightarrow R $ ($R$ is a set of the real number) These functions satisfy ...
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3answers
147 views

$f$ be a smooth function on real line , $f(0)=0$ , $f(x)>0, \forall x \ne 0$ and any $f^{(n)}(0)=0$ ; is $\sqrt f$ smooth?

Let $f: \mathbb R \to \mathbb R$ be an infinitely differentiable function such that $f(0)=0$ , $f(x)>0 , \forall x \ne 0$ and $f^{(n)}(0)=0$ ( the $n$-th derivative ) $, \forall n \in \mathbb N$ ...
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2answers
96 views

How to prove this statement? (Real analysis)

This might be the basic question in real analysis. A function $f$ is $ C^2 $ function on the closed interval$ [0,1]$ Also the function $ f $ is satisfying $ f(0) = f(1) =0 $ Plus, $\vert f''(x) \...
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1answer
55 views

Some doubts with the sign of a derivative

Good evening to everyone. The derivative is defined in the following order: $$ \frac{d}{dx} f(x)=\frac{{-x^2-x+11}}{\left(x+3\right)^2}e^{2-x}\:$$ for $ x < -3 $ $$\:\frac{d}{dx}f(x)=\frac{{x^2+x-...
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1answer
26 views

What is $\vec{v}(\vec{\gamma}(t))$?

If we got the curve $\vec{\gamma}:[0,1]\rightarrow\mathbb{R^3}$ $$\vec{\gamma}(t) = \left(\! \begin{array}{c} t \\ t^2+1 \\ t \end{array} \!\right) $$ And the vector field $\vec{v}:\mathbb{R^3}\...
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6answers
109 views

find ${dy}/{dx}$ if $x^y + y^x = 1$

Find ${dy}/{dx}$ if $x^y + y^x = 1$. I have no idea how to approach this problem. Can somebody please explain this to me?
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0answers
24 views

(Conceptual_Calculus) Differential Conditions v. Derivative Conditions

I have few questions regarding the reason we learn about the differential conditions in higher dimensions and dealing with multivariable calculus. In the context of optimization (e.g. finding ...
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0answers
22 views

mixed partial derivative of a function

Find second order mixed partial derivative, $\frac{\partial^{2} f}{\partial y \partial x}$ of $$\frac{x \log(y)}{ye^x}$$ I am not able approach this problem. I tried differentiating it wrt $x$ (...
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2answers
32 views

Derivative problem(I think, that is Implicit function theorem)

I have a function: $$F(x,y) = 2x^4 + 3y^3 +5xy$$ And input $x$ and output $y$ we know that this relation $F(x,y) = 10$ confirms. We know, that this happens when x = 1 and y = 1. By small change of ...
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0answers
38 views

Is finding the extreme points of a differentiable function by first derivative always correct?

I came across this question where I was asked to find the local minimum and local maximum of the function $$y=\sec x + 2\ln(|\cos x|),$$ domain of $x$ being $(0,2\pi)-\{\pi/2 , 3\pi/2\}$. I found its ...
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1answer
52 views

What is the difference between derevative w.r.t a vector and directional derivative?

Say we have a scalar-valued function $f: \mathbb R^3 \rightarrow \mathbb R$, such that: $$f(\mathbf x) = \mathbf x^T\mathbf a$$ $\mathbf x$ and $\mathbf a$ are two vectors. The derivative of $f$ ...
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2answers
44 views

Help with derivative of integral function?

How do you differentiate: $\displaystyle f(x)=\int_{a}^{b}e^{x^{2}+t^{2}}dt$ I tried writing $f(x)$ as the difference of the antiderivative of the function $\displaystyle e^{x^{2}+t^{2}}$ and I get $\...
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1answer
38 views

Derivative of narrowband signals

I just read a statement in an article that "for a narrowband signal $u(t), -\frac{d^2(u)}{dt^2} \sim u(t)$". Is this appropriate? Here, $u(t)$ is a transient displacement field and we are talking ...
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2answers
79 views

Partial derivative definition

What is the partial derivative of $$\frac{\partial x}{\partial y}$$ when $x$ and $y$ are a part of a function $f(x,y)$? Using an example of: $$f(x,y) = x+y$$ Given the definition of holding all ...
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0answers
22 views

Maxima and minima nth derivative reasoning

I found a statement somewhere in my notes that if we have a higher order function and lets say we take the nth derivative of it. If n is odd and the result turns out to be any number except zero then ...
0
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1answer
22 views

Differentiation product of functions in multidimensional Analysis

Define $k: \mathbb{R}^d \to \mathbb{R}^{m\times m}$ such that $ k(x)=g(x)f(x)^T$, where $f: \mathbb{R}^d \to \mathbb{R}^m, g: \mathbb{R}^d \to \mathbb{R}^m$ are differentiable functions. Prove that $k$...
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2answers
74 views

How can we say the derivative is exact if the difference quotient has a domain restriction?

I think I've finally been able to voice my confusion when it comes to derivatives and limits. Let's first look at the difference quotient for a function $f(x)=x^2$ $$\lim_{h\to0} \frac{f(x+h)-f(x)}{...
3
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0answers
23 views

Defining derivatives and integrals for hyperoperations > 2

Derivatives and Integrals are continuous generalizations of the Forward Difference and Summation additive operators respectively. We can do the same with multiplication and get multiplicative calculus ...