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

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1answer
31 views

Calculus Question ( Surface area/Volume)

An open box (I.e. no lid) had a square base of side $x$ cm and height $h$ cm. Given that the volume of box is $108$ cm$^3$. a) Show that the surface area in cm$^2$ is given by $A = x^2 + 432/x$. b) ...
2
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1answer
18 views

Incongruencies with derivatives and differencials

I read in Piskunov that the increment $\Delta y$ of a function can be written as: $\Delta y = f'(x) \Delta x + \alpha \Delta x$ And, when ${\Delta x\to 0}$ , $dy=f'(x)dx$ The problem is, doesn't ...
2
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2answers
29 views

Can someone explain how to calculate the third order partial derivative of $f.$

$f(x,y)=\sin(xy).$ I calculated that $ \dfrac{\partial^2f}{\partial x\,\partial y}=\dfrac{\partial^2f}{\partial y\,\partial x}=\cos(xy)-xy\sin(xy)$. I also calculated $$ \frac{\partial^3f}{\partial ...
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2answers
22 views

Find the equation of the line tangent to the curve $y=x^2$ parallel to the line $y=x$

Find the equation of the line tangent to the curve $y=x^2$ parallel to the line $y=x$. Just started A level maths, any help is appreciated.
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1answer
33 views

Finding derivative of squrtx

Why does $\frac{1}{\sqrt{z} +\sqrt x }=\frac{1}{2\sqrt{x}}$? Can anyone explain all steps in layman's terms for limit as $z$ approaches $x$ of $\frac{f(z)-f(x)}{z-x}$ when $z= (x+h)$ and $h= (z-x)$.
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0answers
7 views

Approximation of $C^1$ by $C^1_b$

Can we approximate a function which is $C^1$ with functions that are $C^1$ with bounded first derivative? Thank you in advance.
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1answer
304 views

Infinite Series -: $\psi(s)=\psi(0)+\psi_1(0)s+\psi_2(0)\frac{s^2}{2!}+\psi_3(0)\frac{s^3}{3!}+.+.+ $.

We have a given converging series using derivatives and matrices(Analogue to Taylor's series) $\psi(s)_{3 \times 3}=\psi(0)+\psi_1(0)s+\psi_2(0)\frac{s^2}{2!}+\psi_3(0)\frac{s^3}{3!}+..+.. \tag 1$. ...
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2answers
36 views

the derivative of cos(2x) with the double-angle formula?

So, last minute my teacher posted something saying to study double-angle formulas for our derivative test tomorrow. So in the back of the book it shows three things for $\cos x$ $2 \cos^2 x$ ...
1
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1answer
34 views

Integration by parts with Legendre Functions

I need help deriving $\int_{-l}^l [P_l^m(x)]^2 = \frac{2}{2l+1} \frac{(l+m)!}{(l-m)!}$ for the associated Legendre functions I am supposed to use $P_l^m(x) = (-1)^{-m}\int_{-l}^l ...
0
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1answer
49 views

Derivative of all real x

Find the derivative of the function for all real $x$. $f(x)= (\sin(x^\frac 13)^3$) It also gives a hint saying extra attention needs to be placed on $x = 0$. Getting the basic derivative isn't the ...
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2answers
43 views

Computing the nth-derivative $\frac{d^{n}}{d\lambda^{n}}e^{\lambda x-\frac{\lambda^{2}}{2}t}$

According to wolfram-alpha, $\frac{d^{n}}{d\lambda^{n}}e^{\lambda x-\frac{\lambda^{2}}{2}t}= \frac{(-i)^{n} (-t)^{\frac{n}{2}} }{2^{\frac{n}{2}} }e^{x \lambda-\frac{t \lambda^2}{2}} H_n(\frac{(x-t ...
5
votes
3answers
191 views

Are differentiation and integration continuous functions?

Is differentiation a continuous function from $C^1[a,b] \to C[a,b]$? I think it is but I can't prove it... Would it be possible to prove it using theory about closed sets in $C[a,b]$ and their ...
2
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1answer
35 views

Existence of partial derivative

I know how to compute partial derivatives of functions with more than one variable. But how can i assert that the partial derivatives of a given function exist at a point without computing it? ...
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2answers
47 views

Calculate the derivative of $\sqrt{1+\cot^2(x)}$

$$f(x) = \sqrt{1+\cot^2(x)}$$ How to calculate the derivative $f'(x)$? I've been looking at similar problems in my book and at examples, but I'm having a lot of trouble understanding it still. I'd ...
0
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1answer
23 views

Analyze the graph of a derivative

I have the graph of the derivative of some function: And i need to know: a) The critic values of f. b) The X coordinate of each of points where theres an relative extrema of f. c) An interval of ...
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3answers
44 views

How to find $\frac {dy}{dx}$ at the point

If $y=\dfrac{3x^2}{1-4x}$ I am solving through u/v formula but its not working for me Find $\dfrac{dy}{dx}$ for $x=1$.
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1answer
47 views

Differentials (Higher order)

I am having trouble thinking about $\text{dy}$ and $\text{dx}$, $\text{d}^2\text{y}$ and $\text{dx}^2$ as differentials. I get that you can write $\text{dy}=f'(x)\text{dx}$ and how to derive it but ...
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1answer
69 views

Continuous function with continuous one-sided derivative

Simple example of the absolute value function $x \mapsto |x|$ on $\mathbb{R}$ shows that it is possible for a continuous function to posses both the right-hand and the left-hand side derivatives and ...
2
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4answers
177 views

Intiutive argument that $\exp' = \exp$

Is there any intuitive argument or visual "proof" that $\exp' = \exp$? Suppose you have defined the Euler number $\mathrm{e}$ as limit of the sequence $(a_n)$ where $a_n = \left (1 + \frac{1}{n} ...
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1answer
57 views

“The derivative of a sum is the sum of a derivative”. What?

According to this video at this time: "We're gonna do the chain rule here where the derivative of a sum is the sum of a derivative..." Can anyone explain to me why $$ \frac{\partial}{\partial ...
2
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0answers
44 views

Normalize gradient

I want to minimize a function $f \, : \, \mathbb{R}^{N} \, \longrightarrow \, \mathbb{R}$ (with $N \in \mathbb{N}^{\ast}$. In my problem, $N = 315$). I know that $f$ is differentiable on ...
2
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3answers
57 views

Is $x=0$ an inflection point?

Consider $f(x)=x^{\frac {5}{7}}$, is it $x=0$ an inflection point? $$f'(x)=\frac {5}{7}x^{\frac {-2}{7}}$$ $$f''(x)=\frac {-10}{49}x^{\frac {-9}{7}}$$ As far as I know, the inflection point is the ...
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1answer
35 views

Finding Critical Values of Function

$$f(x)=x^{\frac{5}{11}}\cdot(x-5)^2$$ So far, I have used the product rule and chain rule to get... $$\left(\frac{5}{11}x^{\frac{-6}{11}}\cdot(x-5)^2\right)+\left(x^{\frac{5}{11}}\cdot(2(x-5))\cdot ...
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3answers
51 views

How to find this derivative using difference quotient?

how would i find the derivative of $x^8+12x^5-4x^4+10x^3-6x+5$? I know the answer is $8x^7+60x^4-16x^3+30x^2-6$. but how should i solve it using difference quotient, can someone please show the step ...
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2answers
40 views

Differentiate: $f(\theta) = \frac{\sec \theta} {3 + \sec \theta}$

I got $$\frac{(3+\sec \theta) (\cos\theta) - (\sec \theta) (3+\cos \theta)} { (3+\sec\theta)^2}$$ However the program that I am using says my answer is wrong.
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2answers
41 views

Find the critical numbers for this function

$$f(x)=\frac{5x+7}{x^2+x+1}$$ I found the derivative of the function, which is $$\frac{-5x^2-14x-2}{(x^2+x+1)^2}$$ The problem is that I am not getting the correct values. Apparently the answer is ...
0
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1answer
35 views

Using differentials with volume of a cube

my question is The volume of a cube is increased from 1000 cubic centimeters to 1156 cubic centimeters. Use differentials to determine. the side length of the cube increases by? the surface area ...
4
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6answers
581 views

Problem in the second-derivative symbol.

The second derivative of this symbol according to the rules that we have learned the correct mathematical, I wish to know why this symbol is not used.
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5answers
58 views

Differentiate the following function: $y = \frac{2x^2 + 6\sqrt{x} }{9x}$

My answer is $- \dfrac{1}{9 \sqrt{x} }$, however, the program I am using states that I am wrong. Where have I went wrong?
0
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3answers
41 views

what is the derivative of $3\cos(\cos x)\;?$

what is the derivative of $3\cos(\cos x)\;?$ I think I need to use the chain rule and i believed it to be $3-\sin(\cos x)(-\sin x)$ but this is not the case.
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4answers
64 views

What's pratical use of Derivate function calculus? [duplicate]

I would like to know whats the pratical use of derivate calculus? Or what it means? If you can give some pratical example I'll be grateful. Eg.: I can use an definite integral to know area of a ...
0
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1answer
30 views

MacLaurin of the Third-degree in sin(a*x)*cos(b*x) at given values

Alright so from my understanding MacLaurin is a special case of Taylor Series but at f(0). However my question involves solving the third degree of MacLaurin for $$f(x) = sin(a \times x)\times ...
2
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2answers
96 views

Log base 10 not equal to log while differentiating?

I was looking at sample questions from my textbook and I came across something interesting that I need a little help understanding The question was to find the derivative of: $\log_{10} ...
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0answers
11 views

A basic analysis question on gradient

Let $X : \Omega \to \Bbb R^m$ and $Y : \Omega \to \Bbb R^k$ be two random vectors and let $f_w : \Bbb R^m \to \Bbb R^k$ be a map parameterized by $w \in \Bbb R^d$. Consider the function $$g(w) = ...
0
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1answer
30 views

Term for functions with infinite derivatives [closed]

Functions that include a negative indice such as x-1 or similar have an unlimited number of derivatives, so f'(x), f''(x), and fn(x) exist. Is there a technical term for functions like these? I've ...
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0answers
38 views

Directional derivatives in two directions

How can I take a directional derivative in two directions? I mean,$$D_{xy}f(0,0)$$ Because when I have something like $$D_{x}f(0,0),$$ I just use that my direction is in the x axis, $ \vec ...
0
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3answers
43 views

How to find the gradient of norm square

How to find the gradient of the function $f(x) = ||g(x)||^2$ where $x \in \Bbb R^d$ and $g: \Bbb R^d \to \Bbb R$.
4
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1answer
52 views

Given the following derivatives, find the integrals

Find the derivatives of $\ln(x+\sqrt{x^2+1})$ and $\arcsin(x)$, and use the result to find the integrals of the following functions: $$ \dfrac{1}{ \sqrt{ \pm x^2 \pm a^2 }} $$ $$ ...
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3answers
32 views

Differentiating: quotient rule problem

so I am trying to figure out this problem and for some reason I am getting an extra number from what the answer in the book says it is. This picture shows all the work I did and I re-wrote it to make ...
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2answers
35 views

Tanget to the curve, but point not on curve?

Question is Find the $x$-coordinate of all points on the curve $$y=22x\sin(5x)+55\sqrt{3}x^2+68,\quad \frac\pi{10}<x<\frac{3\pi}{10}$$ where the tangent line passes through the point $P(0,68)$ ...
2
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1answer
44 views

How to get tangent of inverse of curve??

Ok so my question is. Let $ f(x)=(1/7)x^3+21x-1.$ and let y=g(x) be the inverse function of f. Determine all points on the graph of the inverse function g so that the tangent line is perpendicular to ...
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3answers
54 views

Derivative proof

This proof is on derivatives. I have no idea where to even begin.
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2answers
32 views

From which equation of motion was this formula derived from in physics

When solving problems involving projectile motion I use: $\sqrt{2 * \dfrac{\text{height above ground}}{9.8}}$ Eg calculate the time it takes for a bomb to impact if it is travelling 4.9km above ...
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1answer
219 views

Critical Numbers And Extreme Values Of A Piecewise Function?

Please determine the critical numbers and extreme values of the : $$ f(x) = \begin{cases} 2-2x-x^2, & -2\le x \le 0 \\ |x-2|, & 0<x<3 \\ \frac {1}{3}(x-2)^3, & 3\le x \le 4 ...
5
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5answers
425 views

L'Hôpital's as $x$ tends to infinity

I'm searching for the explanation to the limit of: $$ \lim\limits_{x\to\infty} x\, \ln\frac{x+1}{x-1}. $$ I know the answer is 2, but I can't seem to get there. The problem is in my textbook under a ...
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1answer
25 views

Confused by informal math: total differentiation

I'm reading these notes that say: total differentiation gives $$ P=a_LW+a_KR\implies dP=a_LdW+a_KdR+[Wd(a_L)+Rd(a_K)]\tag{i}. $$ Please let me explain the notation: we can think of $R,W$ and $P$ as ...
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1answer
34 views

Derivatives and Integrals of Summations

Im unsure if this is just a stupid question because i have been independently studying this kind of math for about a week, but this has been bothering me lately as i have been exploring some definite ...
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3answers
35 views

Help with Inflexion points of a function

I have this function: $P(x) = x^4 +cx^3 + \frac{x^2}{24}$ and i need to find for which values of c the function has: a) two inflection points b) one inflection point c) does not have any inflection ...
0
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1answer
59 views

Real Analysis Question: derivatives

Let $$f''(x)+p(x)\cdot f(x)=0$$ and $$g''(x)+p(x)\cdot g(x)=0$$ where $a<x<b$. 1 ) Show that $W=f'g-fg'$ is a constant on $(a,b)$. 2 ) Prove: If W$\neq$0 and $f(x_1)=f(x_2)=0$ where $a \lt x_1 ...
4
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1answer
39 views

Infinitely differentiable $f: \mathbb{R} \to \mathbb{R}$ such that for each $n \geq 0$, $f^{(n)}(x)=0$ if and only if $x=0$

I am looking for a function $f:\mathbb{R} \to \mathbb{R}$ that satisfies these properties: i) $f$ is infinitely differentiable. ii) $f$ and all its derivatives should intersect the $x$-axis only at ...