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

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1answer
96 views

calculating the taylor term of an integral

an exercise ask me to calculate the Taylor term at $x = 0$ and degree four. I know how to take a derivative of an integral, but I'm having doubts about this one. The function: $$\int_0^x e^{-t^2} ...
1
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0answers
16 views

directional derivatives for a composite function

G $\in$ $C^1(R^2)$ with $G(1,1)-1\ge G(x,1)-x$ for all $x \in R$ and $G(1,1)\le G(1,y)$ for all $y \in R$ $F(s,t)=G(2st-s+1,2st+s+1)$. I've to found the directional derivative of F in $(0,0)$ ...
1
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2answers
20 views

differentiability of a function f(x,y)

I've of $f(x,y)=\sqrt{x^2+y^2} \sin (2 \arctan {y\over x})$ for $x \ne 0$ and $0$ for $x=0$ The function is continuous in all $R^2$. In the points $(0,y_0)$ with $y_0 \ne 0$ the $\partial x ...
1
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1answer
27 views

First part of the proof that $F^*d\beta=dF^*\beta$

Where has the $dy^j$ gone in the highlighted equation? I would have thought the highlighted equation should be $\displaystyle (F^*dg)(x) = \frac{\partial F^j}{\partial x^i}(x)\frac{\partial ...
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1answer
24 views

Component formula for pullback of one forms

Should it not be $\Big(F'(x)v \Big)^j = \frac{\partial F^j}{\partial x^k }(x)v^k$? Then also how is $F^*dy^j = \frac{\partial F^j}{\partial x^i}dx^i$ derived? I cannot what $\beta_j$ has been ...
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0answers
5 views

application of derivatives for optimal dimensions

A printed poster is to have a total area of 589 square inches with top and bottom margins of 4 inches and side margins of 3 inches. What should be the dimensions of the poster so that the printed area ...
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1answer
26 views

Derivative of a scalar product with itself using distributive law

We know that $\frac{\partial (\hat{a}.\hat{b})}{\partial (\hat{a}.\hat{b})} = 1$ But, if we use distributive law $\frac{\partial (\hat{a}.\hat{b})}{\partial (\hat{a}.\hat{b})} = \frac{\partial ...
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0answers
27 views

Another type of primitive

Let $\mathbf{v}:(a,b)\to\mathbb{R}^2$ be a continuous function, such that $\Vert\mathbf{v}(t)\Vert=1,\ \forall t\in (a,b)$. Find all continuous functions $\mathbf{r}\colon (a,b)\to\mathbb{R}^2$ so ...
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1answer
27 views

A function property to guarantee that being constant on an interval implies identically constant

Let $f:\mathbb R\rightarrow \mathbb R$. Suppose we know that $f $ is a constant on some open/closed interval. Which condition does guarantee that $f $ is constant on $\mathbb R$? Clearly, continuity ...
3
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4answers
86 views

Simplifying the derivative of $f(x)= \frac{e^x - e^{-x}}{e^x+e^{-x}}$

I was having some trouble on simplifying the derivative because I didn't know if it's correct. The original function is $$f(x)= \frac{e^x - e^{-x}}{e^x+e^{-x}}$$ What would the simplified derivative ...
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0answers
18 views

Linear approximation with different modifiers

The given function was $$f(x)=ln(\frac{2}{x})$$ and I had to compute the linear approximation at x = 2. I obtained the answer of $$L(x)=-\frac{1}{2}(x-2)$$ I am then supposed to use that ...
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2answers
38 views

Is the Hessian Equal to the Outer Product of the Score with Itself

Recall for a function $f: \mathbb{R}^n \to \mathbb{R}$, the Gradient is \begin{equation} \nabla f(\mathbf{x}) = \begin{bmatrix} \frac{\partial f(\mathbf{x})}{\partial x_1} & \frac{\partial ...
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0answers
8 views

Find the points of inflection and discuss the concavity of the graph?

So I have a few questions about this f(x) = (x-2)^3(x-1) I know the first derivative is (X-2)^2(4x-5) I know the second derivative is 6(2x^2-7x+6) Thusly I also know that the points of inflection ...
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1answer
20 views

Trying to recreate a solution - ODE

I'm trying to copy the method from this video https://www.youtube.com/watch?v=FTHLz2vgj2I to solve the following differential equation: $\frac{dr}{d\theta}+r\tan \theta=\frac{1}{\cos \theta}$ ...
1
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1answer
25 views

Another type of derivative, another type of differential equation

Let $\mathbf{v}:(a,b)\to\mathbb{R}^2$ be a continuous function, such that $||\mathbf{v}(t)||=1,\ \forall t\in (a,b)$. Is it possible to find a continuous function $\mathbf{r}:(a,b)\to\mathbb{R}^2$ so ...
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0answers
12 views

About a several variables differentiable map

I was thinking on this for a while now: let $f:\mathbb{R}^m\to\mathbb{R}^m$ differentiable mapping such that $\|f(x)-f(y)\|\geq C\|x-y\|$, for a certain $C>0$, then show $f$ is an homeomorphism. ...
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0answers
18 views

directional derivatives and differentiability for f(x,y)

If I've a function f(x,y) and is not differentiable at a point then it admits directional derivatives at that point?
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2answers
49 views

directional derivatives in (0,0)

I've this function : f(x,y): $(1+x^2)x^2y^4 \over (x^4+2x^2y^4+y^8)$ for $(x,y)\ne (0,0)$ and $0$ for $(x,y)=(0,0)$ It's admits directional derivatives at the origin?
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0answers
22 views

How can I prove every differentiable funcion is locally lipschitz?

If I have a function $f: \mathbb{R}^m \rightarrow \mathbb{R}^n $ of class $C¹$, how can I prove $f$ is locally lipschitz?
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2answers
51 views

What are the inflection points and concavity of $\;f(x) =3\cos^2(x) - 6\sin(x)$ [closed]

Find the inflection points and concavity of the graph of $$f(x) = 3\cos^2(x) - 6\sin(x), \quad x\in [0, 2\pi]$$
0
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1answer
17 views

Order of derivatives, moving terms past operators

I'm having trouble understanding the following progression of equalities. $\begin{align*} \ddot{x} &= \frac{dv}{dt}\\ &= \frac{dv}{dx} \frac{dx}{dt}\\ &= v \frac{dv}{dx} \tag{1}\\ &= ...
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2answers
29 views

Derivative notation?

I am getting a bit confused with the primed notation for derivatives, does $$f'(g(x))$$ mean the first derivative of $f$ with respect to the spacial coordinate $x$ or with respect to $g(x)$. If it is ...
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0answers
68 views

What's the minimal structure needed to define a notion of derivative?

I know that, for example, to define a limit all you need is the notion of "closeness" generated by a topology; and to define an integral you need a measure function and a sigma-algebra on which it is ...
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3answers
53 views

When finding the derivative using its definition, why do we plug $0$ for $h$?

If $\lim h\to 0$, when finding the derivative of the function, why do you plug in the limit that is being approached. Like why would you plug in $0$ in the function $4x+2h$ (which is the derivative of ...
2
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1answer
97 views

Surjective differentiable function from R to R²

How can I prove there is no function $f: \mathbb{R} \rightarrow \mathbb{R}²$ of class $C¹$ that is surjective? This is an exercise from Analysis on Manifolds, from Munkres. The exercise gives a hint ...
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2answers
32 views

why the derivative of a quadratic function in a linear function?

in a quadratic function, the gradient follows a linear function, as shown by the first derivative. But I cannot accept this, because the rate of change of the gradient seems to be larger when the ...
2
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0answers
49 views

Continuity and differentiability of two variables function

Let be $f:\mathbb{R^2}\rightarrow\mathbb{R}$ defined by: $$f(x,y)= \begin{cases} x^3\log{\left(1+\frac{|y|^\alpha}{x^4}\right)} & \text{if } x \neq 0 \\ 0 & \text{if } x =0 \end{cases}$$ ...
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1answer
13 views

Third derivative of R(t).

How do I do third derivative of the following expression: $R(t) = 7sin(at)\hat x +4e^{-8t}\hat y + 8t^3\hat z$ $(a)$ represents acceleration my goal is to find what $a$ is equal to when $t=0.27778 ...
0
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2answers
25 views

Function + Differentiation

Given the function $f(x) = ax^3+bx^2+cx+d$. Determine the value of $a$, $b$, $c$, and $d$ knowing that the curves passes through points (-1,2), (2,3) and that the tangents at the points on the curve ...
0
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2answers
59 views

Differential equation for $\arctan\sinh x$ [closed]

Can't solve this question after hundreds of tries. If $y= \arctan(\sinh x)$, show that $$ \frac{d^2y}{dx^2} + \sinh x \left(\frac{dy}{dx}\right)^2 = 0. $$
0
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1answer
31 views

Application of the fundamental theorem of calculus. $F(x)=\int_a^x f(t)(x-t) dt$ evaluate $F''(x)$

Let $f$ be a continuous function on $\Bbb R$ $F(x)=\int_a^x f(t)(x-t) dt$ Evaluate $F''(x)$ I used the fundamental theorem of calculus to attempt this question. My attempt at the question ...
0
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1answer
46 views

differentiating a polynomial $k$ times.

If we have the polynomial $$P(x-a)=f(a)+f'(a)(x-a),$$ I can't get how does differentiating the polynomial $k$ times we obtain ...
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0answers
27 views

confused on a related rates problem

I'm having trouble figuring out where to begin with this problem. " a person walks along a path which is on the diameter of a circular courtyard. A light is fastened on the wall at the mid point of ...
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3answers
48 views

Suppose f is a function such that f'(x) = 1/x for all x > 0. Prove that if f(1) = 0, then f(xy) = f(x) + f(y) for all x,y > 0

Suppose f is a function such that $f'(x) = \frac{1}{x}$ for all $x > 0$. Prove that if $f(1) = 0, then f(xy) = f(x) + f(y)$ for all $x,y > 0$ As stated above, how do i go about doing this ...
1
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1answer
18 views

Differentiation to Find slope if tangent line Implicitly

I have the equation $x^3+y^3-4xy=8$, I need to find the equation for the tangent line at $(2,0)$. When I derived the equation I came up with $y'=(3x^2-4x)/(-3y^2-4y)$ (sorry I don't know how to format ...
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4answers
46 views

Evaluate $\lim_{x \to 0} \frac{\sqrt{1 + \tan x} - \sqrt{1 + \sin x}}{x^3}$

What I attempted thus far: Multiplying by conjugate $$\lim_{x \to 0} \frac{\sqrt{1 + \tan x} - \sqrt{1 + \sin x}}{x^3} \cdot \frac{\sqrt{1 + \tan x} + \sqrt{1 + \sin x}}{\sqrt{1 + \tan x} + \sqrt{1 ...
0
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1answer
29 views

Is this a solution to the ODE - simple ODE question

We are given the first order linear differential equation: $y'-2xy=1$ We have guessed a solution to the ODE: $y=e^{x^2}\int e^{-t^2}dt+e^{x^2}$ And we are asked, is this a valid solution to the ode ...
0
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2answers
37 views

Do there exist non-constant function f, g for which the “naive quotient rule” holds?

A common mistake students make is applying a naive form of the quotient rule to functions of the form f/g, mistakenly applying the product rule and arriving at f'g+g'f for the derivative. What I'm ...
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2answers
45 views

When would you want to model a derivative?

I just read this, and am intrigued. http://formulize.nutonian.com/documentation/eureqa/tutorials/modeling-derivatives/ What kind of model would you have a scatter plot of data points, and want to ...
4
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1answer
41 views

doubt regarding Definition of differentiability .

The definition of a differentiable function is as follows: A function $f:A\to Y$ is said to be differentiable at a $\in A$ if there is a linear map $T\in L(X,Y), $ such that : ...
1
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1answer
36 views

Related Rates - Implicitly Deriving Geometric Formulas for Time?

So the step where you implicit derive the geometric formulas for time seem to be giving me the most trouble. How would this be done for volume of a cylinder $$(v=pi^2h)$$ and area of a triangle ...
0
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1answer
21 views

How to take the integral of a derivative to obtain desired result?

I am aiming for the form of derivative below computed over time that causes its differentiated variable V to go from an initial -.001 and increase to reach 10. I will explain my current calcs below ...
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0answers
17 views

Finding the maximum illumination of the edge of a round table top [duplicate]

A light bulb is placed directly above the center of a round table of radius r units. how high above the table should the light bulb be placed to give the best illumination along the edge of the table? ...
3
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4answers
45 views

Prove that $\frac{d}{dx}(\tan^{-1}(x))=\frac{1}{1+x^2}dx$

Prove that $$\frac{d}{dx}(\tan^{-1}(x))=\frac{1}{1+x^2}$$
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2answers
67 views

What is the difference between $\delta x$ and $dx$

Sometimes I see the $\delta x$ and $dx$ but I don't know exactly what is the difference between them.
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0answers
30 views

Prove that $f''(\xi)=0$ for some $\xi \in (a,b)$

A function $f$ is continuous on $[a,b]$ and $f''(x)$ is finite for every $x \in (a,b)$. If the line segment joining the points $A(a,f(a))$ and $B(b,f(b)$ intersects the graph of $f$ at some point $P$ ...
1
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1answer
22 views

Finding the directional derivative of $f(x,y)$.

I have this problem: Let $f(x,y,z)=xyz$. Find the directional derivative in the direction of the velocity vector of the curve $\gamma(t)=(cos(3t),sin(3t),3t)$ on $t=\frac{\pi}{3}$. Is that the max. ...
0
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1answer
18 views

On which points of $xy=(1-x-y)^2$ is the tangent parallel to the $x$-axis?

On which points of $xy=(1-x-y)^2$ is the tangent parallel to the $x$-axis? All I get is the derivative of the function, as far I know, I set the derivative equals to zero.
4
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1answer
41 views

Prove that $f(c)=\frac12(c-a)(c-b)f''(\xi)$

A function $f:[a,b] \rightarrow \mathbb R$ is continuous on $[a,b]$ and $f''(x)$ exists $\forall x\in (a,b)$. If $a<c<b$ and $f(a)=f(b)=0$, prove that there exists a point $\xi$ in $(a,b)$ such ...
1
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1answer
27 views

Partial derivative of convolution

I have a convolution: $$g(x,\alpha) = \int_D \phi(t)f(x-t,\alpha)dt,$$ where $D$ is compact. I need to calculate $\frac{\partial}{\partial \alpha}g(x,\alpha)$. Under what conditions: ...