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

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Differentiate $f(\mathbf{x})=-\sum_{i=1}^n\log(1-x_i^2) \;\;\mathbf{x} \in \mathbb{R}^n$ with respect to $\mathbf{x}$

We have \begin{equation} f(\mathbf{x})=-\sum_{i=1}^n\log(1-x_i^2) \;\;\mathbf{x} \in \mathbb{R}^n \end{equation} where dom$f \triangleq \{\mathbf{x} \mid 1-x_i^2>0 \; \forall \; x_i \in ...
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1answer
38 views

Find the value of $f'(2)$ where $f(x)=\lim_{n\to\infty}\sum_{n=1}^{n}\arctan(\frac{x}{n(n+1)+x^2})$

Find the value of $f'(2)$ where $$f(x)=\lim_{N\to\infty}\sum_{n=1}^{N}\arctan\left(\frac{x}{n(n+1)+x^2}\right)$$ I could not find $f(x)$ here. I had a feeling that Riemann integral should be used to ...
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1answer
38 views

Nearest point to origin of a hyperplane by Lagrange multiplier

Find the point on the hyperplane $x^Tc = β $that is closest to the origin by Lagrange multiplier method. What is hyperplane and how we obtain its origin ! I need a serious hint !!
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1answer
56 views

Fixed Points of Polynomial (Application of Mean Value and Intermediate Value Theorems)

The question is: A number $a$ is called a fixed point of a function $f$ if $f(a)=a$. Consider the function $f(x)=x^{87}+4x+2, x\in\Bbb R.$ (a) Use the Mean Value Theorem to show that $f(x)$ ...
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1answer
54 views

Related Rates - Ships

Ship A is currently 85 km south of ship B. Ship A travels north at 30 km/h and ship B travels east at 20 km/h. How fast is the distance between them changing in 1.5 hours? I have established the ...
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2answers
39 views

Derivative applications

I'm a little bit unsure of how to do these and I'm hoping someone can help me out. The dimensions of a rectangle are changing in such a way that the perimeter remains 24 inches. Show that when the ...
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2answers
68 views

Implicit differentiation to find property of a function

Im asked to show that if $h(1)=0$ and $h'(x)={1\over{x}}$ then $a,b>0$ show $h(ab)=h(a)+h(b)$. Im expected to use implicit differentiation to show this property.
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1answer
36 views

Inverse functions and derivatives

Suppose that $f:[a.b]\to[c,d]$ is differentiable and onto. If $f'$ is never 0 on $[a,b]$ and $d-c\geq2$, prove that for every $x\in[c,d]$, there exist $x_1\in[a,b]$ and $x_2\in[c,d]$ such that ...
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1answer
69 views

Derivative of $\int \limits _x^{x^2} f(t) dt$ with respect to $x$

Suppose that $f$ is continuous on $\Bbb R$. Denote $$G(x) := \int \limits _x^{x^2} f(t) dt$$ Calculate $G'(x)$. Check your expression for $G'$ works for $f(t) \equiv 1$ I've started with $G'(x) ...
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1answer
32 views

Derivative of Heaviside Function and Equivalence

The derivative of the Heaviside function $\theta(x - a)$ is normally taken to be the delta function $\delta(x - a)$. This question has two parts, the first is whether a constant coefficient is ...
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2answers
50 views

To find the $n$th derivative of this function.

Let $f(x)$ be smooth and continuous for $|x|<1$. I am interested in the $n$th derivative of: $$g(x) = f(x) e^{af(x)}$$ for some $a>0$. Is it possible to write this in a neat form? Thanks.
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4answers
77 views

if $F(x)=\ln{x}\ln{(1-x)}$ prove $ F'(x)>0$

an anyone please help me with the following proof: Let $$F(x)=\ln{x}\ln{(1-x)},0<x\le\dfrac{1}{2}$$ show that $$F'(x)>0$$ because $$F'(x)=\dfrac{(1-x)\ln{(1-x)}-x\ln{x}}{x(1-x)}$$ It ...
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4answers
46 views

Derivation of a function over $\frac{1}{\sinh(t)}\frac{d }{dt}$

I can not calculate the next derivative, someone has an idea $$\left( \frac{1}{\sinh(t)}\frac{d }{dt} \right)^n \left( e^{z t} \right)$$ Where $n\in \mathbb N$, $t>0$ and $z\in \mathbb C$. Thanks ...
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4answers
43 views

Derivative of matrices product

Find the derivative of the following matrix $ f(X) = a^TXb, $ where $ a,b ∈ R^n $ and X is an n×n matrix. Please give me some serious hint!
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1answer
58 views

Is the (anti)derivative of an even complex valued function odd, and vice versa?

I'm not sure how much content I can put for a relatively straightforward question, haha. But, I'm attempting to prove something about even and odd functions and integrals in Complex Analysis, and I ...
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1answer
21 views

Derivates of a vector in respect to the elements

Find the derivative of $(a) f(x)= \frac{1}{x_3} \begin{bmatrix} x_1 \\ x_2 \end{bmatrix}$ $(b) f(x)=trace(xx^T)$ where $ x_1 , x_2, x_3 $ are the first three elements of x.
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2answers
38 views

Getting the derivative of the inverse of a function

Given $f(x)$, how would I find $(f^{-1})'(x)$? As an example how would I find that for this problem: $f(x) = 4x^3 + 5x + 2$
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2answers
85 views

Finding extrema of $\frac{\sin (x) \sin (y)}{x y}$

I need to find the extrema of the following function in the range $-2\pi$ and $2\pi$ for both $x$ and $y$, but I don't know how to go about doing it since it's a bit weird and not similar to other ...
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1answer
27 views

Derivative of a nonsingular matrix

Show that : $$\frac{d}{dt} A^{-1}(t) = -A^{-1}(t) (\frac{d}{dt} A(t) ) A^{-1}(t) $$ A(t) is a matrix.
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1answer
38 views

Implicit differentiation to find derivatives of a function whose only defined by its derivative

A question I have asks if $K(x)$ satisfies $K(1)=0$ and $K'(x)={1\over{x}}$ then show: If $f(x)=K(10x)$ then $f'(x)={1\over{x}}$ So Im not sure if I have to prove it or do something else but this is ...
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2answers
39 views

Differentiating an implicit equation.

Google kind of failed me (since I really don't know how to ask it properly) in finding the answer to this question: how do I go about differentiating something like: $$x^2y + xy^2 - y = 1$$ That is, ...
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0answers
30 views

Differentiable iff unique subgradient

Let $f:\mathbb{R}^n\to\mathbb{R}\cup \{ \infty \}$ be convex, $D:=Dom(f)$, $x\in D^o$. Then f is differentiable at x iff $\partial f(x)=\{ d \}$ (a singleton), in which case $\nabla f(x)=d$. ...
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1answer
28 views

Sketch $f(x)=\sin x+\frac{1}{x}$ and local maxima and minima, intervals of increase and decrease,

Sketch $f(x)=\sin x+\frac{1}{x}$ finding local maxima and minima, intervals of increase and decrease. I'm trying to use differentiation to draw this picture and find critical points. So, I get ...
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1answer
60 views

Prove E(x)=pn by taking derivative of Newtons binomial theroem

I have this formula: $E(X)=$$\sum_{k=1}^n k$$n \choose k$$p^k$$(1-p)^{n-k}$ and I am trying to prove $E=(X)=pn$, by taking the derivative of with respect to y: $(x+y)^n=$$\sum_{k=0}^n$$n \choose ...
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1answer
16 views

$\frac{d}{dS} \int_{-\infty}^{d_1(S)} e^{-u^2/2} du = ?$

$\displaystyle{\frac{d}{dS} \int_{-\infty}^{d_1(S)} e^{-u^2/2} du}$ Would the above equate to $e^{-d_1^2/2} \frac{d(d_1)}{dS}$? Why do we include the derivative of the bound? I suppose this is ...
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1answer
161 views

Prove $f(x) = x^{2}\cos(1/x)$ is differentiable [duplicate]

How would I prove that the function $f: \Bbb R \to \Bbb R$, where $f(x) = \begin{cases} x^2\cos(1/x) & x \neq 0 \\ 0 & x=0 \end{cases}$ is differentiable? So far I have tried using the ...
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1answer
82 views

Step by step solution to these derivations [closed]

I will write test tomorow and I'm stuck at few derivation examples. Unfortunatelly, teacher don't gave me a solution to these examples. I have no problem with simpler derivations. Calculate first ...
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2answers
50 views

Limit of infinity times 0

I have a question regarding a specific step in the proof of the theorem that 'differentiability implies continuity'. The proof in my calculus book asserts that if $h\to0$ then: ...
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1answer
32 views

When does the minimum of: $\int_0^1 f^2(x) dx - 2\lambda \int_0^1 x f(x) dx + \frac{\lambda^2}{3}$ occur?

When does the minimum of: $\displaystyle{\int_0^1 f^2(x) dx - 2\lambda \int_0^1 x f(x) dx + \frac{\lambda^2}{3}}$ occur? I have no clue, other than this looks like a quadratic.
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2answers
59 views

Derivative of piece-wise function given by $x\sin\frac1x$ at $x=0$

Given the function: $$f(x) = \begin{cases} x\cdot\sin(\frac{1}{x}) & \text{if $x\ne0$} \\ 0 & \text{if $x=0$} \end{cases}$$ Question 1: Is $f(x)$ continuous at $x=0$? Question 2: What is the ...
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2answers
74 views

Notation for higher degree derivatives

Lebniz's notation for ordinary derivatives as quotients of differentials is a convenient abuse of notation, since it lets you express things like the chain rule and the derivative of the inverse ...
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0answers
27 views

Determine saddle point if second derivative test fails

I have the following function $f(x, y) = x^3 + y^3$, which, by looking at the 3D plot, I can figure out there is a saddle point at $(0, 0, 0)$. However, the second derivative test fails with the ...
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1answer
86 views

How to get nth derivative of $\arcsin x$

I want to calculate the nth derivative of $\arcsin x$. I know $$ \frac{d}{dx}\arcsin x=\frac1{\sqrt{1-x^2}} $$ And $$ \frac{d^n}{dx^n} \frac1{\sqrt{1-x^2}} = \frac{d}{dx} (P_{n-1}(x) ...
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1answer
65 views

Rectangular box without a top has a volume of 216 in$^3$. Find dimensions of the box with the smallest surface area. [closed]

Rectangular box without a top has a volume of 216 in$^3$. Find dimensions of the box with the smallest surface area. Use second derivative test. So this is what I came up with not sure where it ...
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1answer
53 views

Does the existence of the derivative at a point imply the existence of the left and right derivative?

I'm asking this because I've seen "results" in the internet that state: A function $f$ is differentiable at $x = a$ if and only if both the right-hand derivative and left-hand derivative at $x = a$ ...
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2answers
78 views

derivative on both sides

I am reading about feedback topologies and having some problems about math. I am not a math student so I need your help. Could you explain if the operation of taking derivative of both sides correct ...
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1answer
45 views

Find series of real functions for which the sum has continuity properties

The sequence of continuous real functions $f_i$ is defined on the unit interval $[0, 1]$. Each $f_i$ is composed of finitely many linear segments, each segment has slope +1 or −1, moreover $f =\sum ...
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0answers
56 views

Partial Derivatives Calculus

If $$x^x\cdot y^y\cdot z^z=c$$ then prove that $$\frac{\partial^2z}{\partial x \, \partial y}=(-x\log_ex)^{-1}$$ How to do this? I was first taking logarithm and then differentiating but did not work. ...
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1answer
49 views

When there exists function $f$ such that for given $g$ we have $f'=g$?

I am looking for a theorem that states when function $g: \mathbb R \mapsto \mathbb R$ is a derrivative, i.e. there exists $f$ such that $f'=g$. What about if we just need this condition almost ...
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5answers
161 views

Why this function is continuous and not differentiable at point $x=1$

I have a function $$f(x) = \begin{cases}x^2+2,& x\leq 1\\x+2 ,& x > 1\end{cases}$$ I have to show that this function is continuous and not differentiable at point $x=1$, but when I look for ...
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2answers
52 views

$n$-th derivative of $f(\ln x)$

Find general formula for $n$-th derivative of $y = f(\ln x)$. To start with I found couple of derrivatives: \begin{align} y' &={1 \over x}f'(\ln x) \\ y'' &={1 \over x^2}(f''(\ln x)-f'(\ln ...
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1answer
31 views

Proving differentiability on every interval equals proving differentiability on all of $\mathbb{R}$?

The question actually came to me from series of functions. Suppose $f_n(x)$ is a sequence of continuous differentiable functions, that $$\sum_{n=1}^\infty f_n'(x)$$ converges uniformly on any closed ...
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1answer
62 views

Application of Euler Theorem On homogeneous function in two variables.

Euler theorem says, If $$u=f(x,y)\text{ ,homogeneous}$$ Then,$$x\frac{\partial u}{\partial x}+y\frac{\partial u}{\partial y}=nu$$ Where $$n\to \text{degree of function}$$ Question If ...
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3answers
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Definition of derivative $f(x) = 3x - \frac{1}{x^2}$

$f(x) = 3x - \frac{1}{x^2}$ I am finding this problem to be very tricky:
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2answers
69 views

Find the derivative for $f(x) = \sqrt{x+2}$ by definition

Use the definition of derivatives to show that the derivative of $f(x) = \sqrt{x+2}$ is $$ f'(x) = \frac12(x+2)^{-\frac12} $$ My proof so far: Given $\epsilon > 0$ , $\exists ...
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1answer
49 views

Total derivative and chain rule

I am aware this could be a dumb question but I got myself stuck in some complicated calculus that I have greatly simplified to focus on my lack of understanding. My goal is to compute the derivative ...
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0answers
31 views

Find for which values of C a function is differentiable

Find for which values of $C \in \mathbb{R}$ the function $f$: $\mathbb{R^2} \to \mathbb{R}$ is differentiable, with $f$ defined by: $$f(x,y) = \begin{cases} \frac{|x|^C y}{\sqrt{x^2 +y^2}} ...
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1answer
29 views

Derive a power series expansion about $x=0$

$(1+x)^2y''(x) - xy'(x) + y(x) = e^{-x} $ If possible, derive a power series expansion about $x=0$ up to the first 3 terms. State which parts are the particular solution and which parts are the ...
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1answer
28 views

Computing the derivative: $\frac{\partial}{\partial x} \left\{ \int_0^t \int_{x - t + \eta}^{x + t - \eta} F(\xi,\eta) \,d\xi\, d\eta \right\}$

Let's say that $F$ is a nice well-behaved function. How would I compute the following derivative? $$\frac{\partial}{\partial x} \left\{ \int\limits_0^t \int\limits_{x - t + \eta}^{x + t - \eta} ...
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1answer
18 views

How do i calculate the number of subintervals n in Midpoint method?

I want to calculate the least error (o) in order to obtain the exact answer for integration using the midpoint method. However I am having trouble doing so since i was given a functions whose second ...