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

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1answer
275 views

How to restrict Lagrange multiplier on positive values?

Here's the function that i want to optimize: $$f(x,y) = x-2y$$ and the constraint is: $$g(x,y) = x^2 + y - 10 = 0$$ Solving with Lagrange multiplier I get: $$F(x,y) = x-2y - x^2\lambda - y\lambda ...
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3answers
137 views

Differentiable function, with $f'(x)=[f(x)]^2$

Let $f: \mathbb{R} \to \mathbb{R}$ be a differentiable function such that $f(0)=0$ and $\forall x \in \mathbb{R}$, we have $f'(x)=[f(x)]^2$. Show that $f(x)=0, \forall x \in \mathbb{R}$.
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0answers
247 views

Derivative Riccati-Bessel function

I have found two derivatives of the so-called Riccati-Bessel functions in a textbook $$ (x j_n(x))'=xj_{n-1}(x)-nj_{n}(x)$$ and $$ (x h_n^{(1)}(x))'=x h_{n-1}^{(1)}(x)-n h_n^{(1)}(x)$$ so $j_n$ is ...
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4answers
245 views

Differentiate $\ln(\cos2x)$ With respect to $x$.

I need to differentiate $\,\ln(\cos2x)$. Can someone please explain how to do this question? Thank you.
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2answers
143 views

Differential of normal distribution

Let $$f(x)=\frac{\exp\left(-\frac{x^2}{2\sigma^2}\right)}{\sigma\sqrt{2\pi}}$$ (Normal distribution curve) Where $\sigma$ is constant. Is my derivative correct and can it be simplified further? ...
0
votes
1answer
117 views

Trigonometric Functions. Definite Integrals

Find, correct to one decimal place, the value of $$\int_{0}^{60} 2\sin(x/2) \, dx.$$ Can someone please show me how this question is done. It would be very helpful thanks!
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3answers
277 views

A question on generalization of the concept of derivative

I am looking for some material to understand the process of generalization of the concept of derivative. I would not like to just read and apply the definition of the concept of differentiation in ...
1
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1answer
26k views

Slope of line tangent to a curve at a given point, using First Principles

Find, from first principles, the gradient of the tangent to the curve $y = 5 - x^2$ at the point $(1,4)$ on the curve. So I'm currently lost on this question can some one please show me the solutions ...
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2answers
82 views

Why do we have $\psi^{(m)} (z)=(-1)^{m+1}\int_{0}^{\infty}\frac{t^me^{-zt}}{1-e^{-t}}dt$?

Why is the following representation true? $$\psi^{(m)} (z)=(-1)^{m+1}\int_{0}^{\infty}\frac{t^me^{-zt}}{1-e^{-t}}dt,$$ where $\psi^{(m)} (z)$ denotes the Polygamma function.
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1answer
74 views

Taylor Expansion with Integral Remainder Question

I have the following question at hand and I have to admit that I am not used to integral remainder form of taylor approximation. I am still trying to work around, so a couple of hints would be useful ...
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2answers
70 views

Finding x when slope = 1

I've been working out some problems relating to slope on the points of a curve. I'm having issues with this one: In the curve to which the equation is... $$x^2 + y^2 = 4$$ find the value of $x$ at ...
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2answers
88 views

Simplifying second derivative using trigonometric identities

Given that $x=1+\sin(t)$ , $y=\sin(t) -\frac{1}{2} \cos(2t)$ show that $\frac{\text{d}^2y}{\text{d}x^2}=2$. I am having trouble proving this. Here is my working so far: ...
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2answers
46 views

Derivative problem

Could anyone help me? Let $f:[0,1] \to \mathbb{R}$ be a function of class $C^1$ such that $f(0)=0$ and there exists $a \in ]0,1[$ with $f(a)f’(a)<0$. Show that there exists $b\in ]0,1[$ with ...
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1answer
141 views

Showing limit of a derivative is finite

Given that a function $f$ is continuous on interval$\left[a, b\right]$, and that its derivative is finite everywhere on that interval except possibly at $c$. I am also given that $lim_{x \rightarrow ...
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3answers
71 views

Is there any function that never gives an answer other than 0/0 when applying L'Hôpital's rule?

Someone asked this question in my calculus class and the teacher said that he would get back to the student on that one. I never heard back, so was hoping someone here knew the answer? EDIT Sorry ...
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3answers
112 views

continuity of the derivative under certain conditions

I am working on this exercise in a book which asks to prove that $f$ is differentiable if $f$ is continuous and that $\lim \limits_{x\rightarrow x_0} f'(x)$ exists. I know that this is easy to show ...
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0answers
60 views

Supremum and Infimum of a complex expression

I read something on integral transforms, one problem forces me to have to solve the following problem: Let $a,b$ and $p$ be positive numbers so that $a<b$ and $p\geq2$. Find the supremum and ...
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3answers
100 views

For what values will f(x) be necessarily one-one?

Let $g:\mathbb{R}\to\mathbb{R}$ be a differentiable function such that $|g'(x)|\le M$ for all $x\in \mathbb{R}$. For what values of $\epsilon$ will the function $f(x)=x+\epsilon g(x)$ will be ...
1
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2answers
51 views

Slope of a points in a Circumference

I need to compute the slope (m) of the nine points of a circumference divided in equal parts. But a circumference is not a function. I make a Geometric approach but I am not satisfied with it. Do ...
2
votes
1answer
67 views

Show $\partial _x \int_{(x_0, y_0)}^{(x,y)}P(s,t)ds + Q(s,t)dt = P(x,y)$

There is a theorem from advanced calculus that I'm trying to prove. Suppose $P(x,y)$, $Q(x,y) \in C^2$ on a simply connected domain $D$, and suppose that $P_y = Q_x$ (i.e. $\omega = Pdx + Qdy$ is ...
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2answers
417 views

Derivative of Standard Normal Inverse

How can I calculate the derivative of the standard normal inverse. I think the derivative of $\Phi^{-1}(x)$ is $$\frac{1}{\phi(\Phi^{-1}(x))}.$$ I would like to know how to find the derivative of ...
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2answers
133 views

Use Lagrange multipliers to find minimum and maximum

$$f(x,y,z) = x^{2}y^{2}z^{2}$$ If: $$g(x,y,z)=x^{2}+y^{2}+z^{2}+1 = 0$$ The method I know is to create the following function: $F(x,y,z,\lambda)=f(x,y,z)-\lambda g(x,y,z)$ Then create system of ...
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1answer
50 views

Deriving the Pratt-Arrow Definition

I am looking through the definition of Pratt-Arrow Definition and I saw this equation. I am not sure how this derivation comes about. $$\begin{align*} \mathrm{E}[U(W+\epsilon)] &\simeq ...
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0answers
83 views

Rate of change of second order derivative -Third Derivative

I'm familiar with the standard time-derivative interpretations in which the first time derivative of a position function is velocity, the second derivative is acceleration, and the third derivative is ...
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1answer
106 views

Derivative of $f(x) = a \sum_{i=1}^x 10^{x-i}$

Consider the following function: $f(x) = a \sum_{i=1}^x 10^{x-i}= a (10^{x-1} + 10^{x-2} + \cdots + 10^{x-x})$ whose domain are the positive integers greater than 1. What is its derivative ...
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1answer
205 views

Higher Derivatives of trigonometric functions

The position of a particle is given by $s = 5 \cos (2t+ (\pi/4))$ at time $t$ . What are the maximum values of the displacement,the velocity,and the acceleration? The answers are displacement: $5$ ...
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2answers
2k views

Derivative of an integral $\sqrt{t}\sin t dt$

I need to find the derivative of this function. I know I need to separate the integrals into two and use the chain rule but I am stuck. $$y=\int_\sqrt{x}^{x^3}\sqrt{t}\sin t~dt~.$$ Thanks in advance ...
0
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1answer
204 views

Material derivative of a material vector field

On page 12 of An Introduction to Theoretical Fluid Dynamics, following the introduction of a material vector field $v_i(\mathbf a,t)=J_{ij}(\mathbf a,t)V_j(\mathbf a)$ the author wrote: $$ ...
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2answers
95 views

Differentiation, an issue with an exercise

I'm currently working on an exercise that involves quite a few fractional exponents. This is it: $$y = \frac {(x^4 + a)^\frac {1}{3}} {(x^3 + a)^ \frac {1}{2}} $$ I take the multiplication route by ...
4
votes
1answer
116 views

Is this really a typo?

Let $U \subseteq \mathbb R^n$ and $F: U \to \mathbb R^m$ a function with coordinate functions $f_i$. My notes say that: If $F$ is differentiable on $U$ the Jacobian of $F$ is defined at each point in ...
4
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2answers
187 views

Formal (series/sum/derivative…)

I have come across a lot of cases where terms such as formal sum rather than simply sum is used, similarly in case of derivatives/infinite series/power series. As I understand in case of series/sum, ...
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1answer
130 views

Judging function based on its first and second derivatives

I wish somebody could help me with this. Let $f$ be a twice differentiable function on $\mathbb{R}$. Given that $f′′(x)>0$ for all $x \in \mathbb{R}$. Then which of the following is true? ...
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2answers
56 views

how to calculate partial derivative?

how do I find $\frac{\partial q}{\partial k}$ of $q(k,l,m) = k\,p(k,l) + m^2$ ? I have tried $\frac{\partial q}{\partial k}= p(k,l) \times\begin{bmatrix}\frac{dk}{dm}+\frac{dl}{dm}\end{bmatrix} + ...
5
votes
1answer
155 views

Differentiation under the integral sign??

When I tried to show this, I didn't get the integral of the derivative - only the other terms, and I have no idea why. Here's the working I have; $\dfrac{d}{dx} \displaystyle\int_{a(x)}^{b(x)} ...
2
votes
1answer
92 views

Derivative of the inverse of $y=(a+bx)e^{cx}$

I need to solve for the 1st derivative of the inverse of $y=(a+bx)e^{cx}$ but my calculus is a bit rusty. I know that to get the inverse function, I would have to use the Lambert W method but I think ...
0
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3answers
255 views

Differentiate $y=\tan^3 (5x+4)$?

Differentiate $$y= \tan^3 (5x+4).$$ So I know we have to use the product rule, but since I haven't done this for a while, can someone please show me how? THANKS!
2
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2answers
40 views

How to prove that $f$ is $1-1$ from $E$ on $\{ (s,t) : s> 2\sqrt{t} >0\}$

Question: Let $E=\{(x,y): 0<y<x \}$ set $f(x,y)=(x+y, xy)$ for $(x,y)\in E$ a) How to prove that $f$ is $1-1$ from $E$ on $\{ (s,t) : s> 2\sqrt{t} >0\}$ And how to find formula for ...
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1answer
92 views

First order partial derivatives

Suppose that $f:\Bbb R^2\to \Bbb R^2$ has $C^1$ partial derivatives in some ball $B_r(x_0,y_0)$ $r>0$. Prove that if $\Delta_f(x_0,y_0)\neq 0$, then $\displaystyle\frac{\partial f_1^{-1}}{\partial ...
3
votes
1answer
32 views

IFT application.

Suppose that $f:=(u,v):\Bbb R\to \Bbb R^2$ is $C^2$ and $(x_0,y_0)=f(t_0)$ A) prove that if $f'(t_0)\not=0$ then $u'(t_0)$ and $v'(t_0)$cannot both be zero. B) if $f'(t_0)\not=0$ show that either ...
3
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1answer
103 views

$(a,b)$ is saddle point if $f_{xy}(a,b)\not= 0$

Suppose that $V$ is open in $\Bbb R^2$ that $(a,b)\in V$$\ \ \ \ \ f:V\to\Bbb R$ has second order partial total differential on $V$ with $f_x(a,b)=f_y(a,b)=0$ If the second order partial derivatives ...
3
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1answer
47 views

A doubt over a differentiation problem.

The question is: Find the equation of the tangent to the curve $y^2=8x$ at a point $(x_0,y_0)$. My teacher's approach : Differentiate the equation. we get $2y\cdot \dfrac {dy}{dx}=8 $ ...
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1answer
129 views

If $f<1$, $f(0)^2 + f'(0)^2=4$, exists $x_0$ s.t. $f''(x_0) + f(x_0)=0$

Suppose $f:\mathbb{R}\to\mathbb{R}$ is $C^2$, $f < 1$ for all $x$, and $f(0)^2 + f'(0)^2=4$. Show that $\exists x_0$ s.t. $f''(x_0) + f(x_0)=0$. So far, I have let $\phi(x) = f(x)^2 + f'(x)^2$. ...
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2answers
58 views

Differentiation of $x$ to the power of $y$ with respect to $x$

As the title suggests, I need to differentiate $x$ to the power of $y$ with respect to $x$. Not sure how to start. Do I need to take natural log on both sides? That is: $\dfrac{d}{dx}x^y=?$
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1answer
53 views

General Derivative for Negative Binomial Probability Generating Function

Find $\cfrac{dh_M^k}{ds}$ for $h_M(s)=p^r (1-qs)^{-r}$ I know from calculus $y=x^n$, then $dy/dx=nx^{n-1}$. So I thought I'd take a few derivatives and see a pattern then generalize: ...
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2answers
99 views

If $\int f(x) \sin{x} \cos{x}\,\mathrm dx = \frac {1}{2(b^2 - a^2)} \log f(x) +c $. Find $f(x)$

Problem: If $\int f(x) \sin{x} \cos{x}\,\mathrm dx = \frac {1}{2(b^2 - a^2)} \log f(x) +c $. Find $f(x)$ Solution: $\int f(x) \sin{x} \cos{x}\,\mathrm dx = \frac {1}{2(b^2 - a^2)} \log ...
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2answers
239 views

Computation of the Wirtinger derivative of a product

Let's have a function $f = (A/2)\phi\bar{\phi}$, where $\phi=\phi(z)$ is a complex-valued scalar field. I need to obtain $df/d\phi$. If I treat the real and imaginary parts of $\phi$ as independent ...
4
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0answers
242 views

Uniform Differentiability

Let $f:\mathbb{R}^n \to \mathbb{R}$ be differentiable and such that $\nabla f$ is uniformly continuous. Show that $f$ is uniformly differentiable; that is, for any $\epsilon >0$, there is a $\delta ...
5
votes
1answer
410 views

Finding the derivative of $\sin \sqrt {x^2+1}$ from the definition?

This means finding $\lim_{h \to 0} \large \large \frac{\sin \sqrt {(x+h)^2+1}-\sin \sqrt {x^2+1}}{h}$ . The only way I could think of to do this is to replace $h$ by some function $f(h)$ such that ...
2
votes
1answer
74 views

Find force required for a launch between two points

Let me start by saying this is within a game environment, so gravity isn't 9.81m/s^2, and the unit of measure for distance will be "blocks". I'm attempting to find the amount of force needed in order ...
1
vote
1answer
138 views

Cauchy Derivative Estimates for entire functions with a bound.

The problem statement: Assume $f$ is an entire function and that there is an $n \in \mathbb{N}$ and a $C < \infty$ such that for $z \in C$ $$|f(z)| \le C ( 1+|z|^n)$$ Also assume that $f$ is never ...