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

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1answer
20 views

Second derivative with implicit differentiation

Question: Determine whether the given relation is an implicit solution to the give differential equation. Assume that the relationship does define y implicitly as a function of x and use implicit ...
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10 views

Proof of Darboux's Theorem when the function has infinite derivatives at both endpoints.

I have a question about the statement in the NOTE above. It says that the Darboux's Theorem is also valid when one or both the one-sided derivatives are infinite. So say $f_{+}'(a)=-\infty, ...
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0answers
17 views

continuity of the piecewise functions [on hold]

$1$. $g(x)=0$,if $x$ is irrational and $g(x)=x$ if $x$ is rational Find all points of at which $f$ is continuous. $2$. Let $A$ and $B$ be compact sets. Define $A+B =$ ...
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0answers
12 views

How to verify that the following function is monotone increasing?

$\displaystyle f(x)=x\cdot\left(1-\frac{C_Bx^{B}}{\sum\limits_{k=0}^{B}C_kx^k}\right)$, where $0<x<1$, $\displaystyle C_k=\binom{n+k}{k}$, $n,B$ are integers, then, how to verify that $f(x)$ ...
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0answers
8 views

Singular points while differentiating a function with respect to another function

I have $z(x) = \frac{df(x)}{dx}$ where $f(x)$ if a function of x. I'd like to have the derivative of $z(x)$ in respect to $f$: $\frac{dz}{df} = \frac{\partial f'(x)}{\partial x} \frac{dx}{df}$ ...
5
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1answer
70 views

A possible dumb question about derivative

I was solving some differentiation problems when I found the function $$g(x)=\sqrt{x+\sqrt{x+\sqrt{x}}}.$$ So I thought: If I define the function $f:\mathbb{R_{x>0}}\to \mathbb{R}$ as ...
2
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3answers
83 views

Proving $\sin^2(x) + \cos^2(x) =1$ using calculus

Ok so the book in which I found this doesn't say mention the trigonometric functions by name but the question is: Let $s(x)$ and $c(x)$ be functions satisfying $s'(x)=c(x)$ and $c'(x)= -s(x)$ for ...
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1answer
23 views

L'Hopital's rule and limiting variables

I'm working some problems from a calculus text and came across this question: If $f(x)$ is a function that's differentiable everywhere, what is the value of the limit $$\lim\limits_{h \to ...
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3answers
27 views

Help with a derivative of integral please.

I'm supposed to calculate the derivative of $\frac{d}{dx}\int_{x^{2}}^{x^{8}}\sqrt{8t}dt$ the answer I got is $8x^7\cdot \sqrt{8x^8}$ but when I put this into the grading computer it is marked wrong. ...
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2answers
26 views

Let $f(x)=\arctan x -\frac{ln|x|}{2}$. Then $f(x)$ is increasing in?

Let $x<0$. $f'(x)=\frac{1}{1+x^2}+\frac{1}{2x}>0$ $$f'(x)=\frac{(1+x)^2}{2x(1+x^2)}$$ which is always negative. But answer is $x\in (-\infty,0)$.
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0answers
16 views

Proofs of the multiplication or chain rule for derivatives that invoke symmetry

Introductory calculus texts sometimes include direct proofs of the multiplication and chain rules for derivatives by: Introducing a pair of differences $D_f=\frac{f(x+h)-f(x)}{h}-f'(x)$ and ...
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1answer
23 views

Lowering the order of a linear differential equation

Let $$L(x) \equiv x^{(n)}+a_1(t)x^{(n-1)}+...+a_{n-1}(t)x'+a_n(t)x=0.$$ and let the following solutions be given: $x_1,x_2,...,x_m(m<n)$- linear independent solutions. Let's find: $x_{m+1}, ...
3
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1answer
44 views

Why do we write $df/dx$ instead of $df/dx(x)$?

I was just thinking about how, i.e., if $f\colon\mathbb R\to\mathbb R$ is defined by $f(x) = x^2$, then it's customary to write $$ \frac{df}{dx} = 2x. $$ But since the derivative is itself a function ...
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0answers
20 views

find the stationary points for $f(x)=x^{\frac 2 3}$.difference between the stationary point and critical point and one more called turning point.

Find the stationary points for $f(x)=x^{\frac 2 3}$. My work I realized the following $\spadesuit$ $f'(x)=\frac 2 3 x^{-\frac 1 3}$ which is not defined at $x=0$ $\spadesuit$ $f'(x)<0$ for ...
5
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2answers
521 views

Only once differentiable

Is there any example of a real function that is one-time-only differentiable, meaning there is $f'(x)$, but no $f''(x)$? I haven't been able to find any example... Of course it would be preferred if f ...
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2answers
55 views

To determine the existence and the value of $\lim_{x \to 0} \frac {2^x-1} x$

I would like to somehow firstly show that $$\lim_{x \to 0} \frac {2^x-1} x$$ exists and determine the value of the limit. My first ideas were by Monotone Convergence. I have been able to prove that ...
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2answers
22 views

The supremum value of $x^{2}y^{2}(x^{2}+y^{2})$ when $x+y=2n$ for some fixed $n\in \mathbb N $

Let $S$ be the set of all tuples $(x,y)$ such that $x+y=2n$ for a fixed $n\in \mathbb N$. Then what is the supremum value of $x^{2}y^{2}(x^{2}+y^{2})$ $?$ I substituted $y=2n-x$ ...
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2answers
67 views

Wolfram Alpha “x = derivative x”

Asking Wolfram Alpha $x = \text{derivative } x$, I was expecting $e^x$, being that the derivative of $e^x$ is $e^x$, Wolfram Alpha however yields $x = 1$. Is this stating that the derivative of a ...
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1answer
31 views

Infimum of $f(x): (0,\infty) \to \mathbb R$, $f(x) = \ln(e^x-1)+\frac2x-x$

Calculate infimum of $f(x): (0,\infty) \to \mathbb R$, $f(x) = \ln(e^x-1)+\frac2x-x$ I calculate derivative $$f'(x)= (\ln(e^x-1)+\frac2x-x)' = \frac1{e^x-1}e^x-\frac2{x^2}-1 = ...
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2answers
30 views

Find all points such that function has all partial derivatives in that point.

Find all points $(x,y) \in \mathbb{R}^2$ such that function has all partial derivatives in that point.$$ f(x,y) = \begin{cases} \frac{\sin(xy^2)}{y} &\mbox{if } y>0 \\ xy^2 & \mbox{if } y ...
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2answers
33 views

Derivative of implicit function with exponential functions of each other

We have the equation: $$ x^y = y^x +y $$ Which defines an implicit function $y(x)$ at the point $(2,1)$. I'm asked to find the derivative at $y'(2)$. I saw the answer in Wolfram: $$ y'(x) = \frac{y ...
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0answers
8 views

How to compute the gradient of the weighted kernel

Let's say $f(X) = \sum_{i,j}A_{i,j}x_i'\cdot x_j $ where $x_i,x_j$ are the i-th, j-th columns of $X$. So what is the gradient $\frac{\partial(f(X))}{\partial{X}}$ ?
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25 views

Find $C^1$ class function such that

Given: $$g: \mathbb{R}^3 \rightarrow \mathbb{R}, g(x,y,z)=z^3-3xyz-x-8$$ Decide whether in the neighbourhood of the point $(x,y)=(0,0)$ there exist $C^1$ class function $z=z(x,y)$, such that ...
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1answer
46 views

Differentiating the definite integral of x*f(x)

I'm trying to differentiate the integral below. I was wondering how I could approach it. z ~ N(a , $b^2$ $\cdot$ $x^{-2}$) $\frac{d}{dx} \int _{-\infty} ^ {z^*(x)} z \cdot \Phi ^{\prime} (z) dz$ ...
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0answers
32 views

How to derive: div ( s * [ grad(U) ]^T )

Having some questions on the derivation of the following equation (seen in someone's PhD thesis): where s is a scalar valued function, ...
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0answers
43 views

tricky derivative with logarithm of sum

I'm having trouble understanding the solution of a limit. It involves a formula for measuring certainty of a discrete probability distribution. Given a set of values $p_j$ which sum up to 1, find the ...
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0answers
31 views

Differentiating CDF

I'm trying to differentiate the cdf of z with respect to x where the upper bound is a function of x and z ~ N(a , $b^2$ $\cdot$ $x^{-2}$) $\frac{d}{dx} \int _{-\infty} ^ {z^*(x)} \Phi ^{\prime} (z) ...
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0answers
11 views

Minimum directional derivative

I know that maximum of $(D_uf)= ||grad f||$ Because $(D_uf)= grad f. u = ||grad f||.||u||.Cos\theta$ So how about the minimum value? I believe it must be $-||gradf||$ since $min(cos\theta = -1 ) ...
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3answers
42 views

Calculus Lab Problem Differentiation [on hold]

I am kind of confused on how to differentiate this function. Here is the problem: $$P(t) = \frac{1}{(1 + e^{-t})} $$ Use the differentiation rules to compute the exact value of $P'(2)$. Certain ...
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1answer
41 views

Differentiation Commute with Lebesgue Integration

My question is simple: Given $f: \mathbb{R}^{n+m} \to \mathbb{R}$, $f\in C^{k}(\mathbb{R}^{n+m})$ , and $X \subset \mathbb{R}^{n}$. Write $f$ as $f(x_1, \ldots, x_n, t_1, \ldots t_m)$. When is ...
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3answers
28 views

Find the area between the given function , and two tangents off of the point (2,-2)

So here is a general graph of the first couple directions. $T_1$ and $T_2$ are supposed to be the points where the tangent line intersects the parabola. The tangent lines and points where the ...
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0answers
22 views

how can I prove that a derivative of an implicit function is bounded?

I have the following implicit function $V(\tau,\mu)$. The function is bounded and continuous and differentiable on $\mathbb{R}$. What other properties or assumptions should I make or what conditions ...
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0answers
22 views

From inequality on derivatives to inequality on functions

What is the set of differentiable functions $f$ that satisfy the following inequalities for all $x\geq 0$: $0\leq f'(x)\leq e^{-x}$ Initially, I thought I should just integrate the inequality and ...
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1answer
19 views

Piecewise $C_1$ and piecewise continuous

I would appreciate if the following questions could be clarified with your help. If a function is piecewise $C_1$, does this imply that it's also piecewise continuous? If a function is piecewise ...
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2answers
22 views

Finding Tangent Line Using Limit Definition

I'm supposed to get the equation of the tangent line to the graph of $f(x)= \frac{8}{x}$ at the point $(2,4)$. I started with $$\frac{\frac{8}{x+h} - \frac{8}{x}}{h},$$ then I cross multiplied: ...
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1answer
19 views

Conditions for convergence of derivatives from pointwise convergence

Let $\{f_n\}_n$ be a sequence of functions $\mathbb R \rightarrow \mathbb R$ which converges pointwise to $f$, ie: $$f_n(x) \rightarrow f(x) \hspace{10pt}\hbox{for all $x$}.$$ What additional ...
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1answer
22 views

Characterizing connected sets of $\mathbb R^n$ is terms of differentiable maps for which zero derivative everywhere implies constant

Let $U $ be an open subset of $ \mathbb R^n$ ; then how to prove that $U$ is connected iff for every differentiable function $f:U \to \mathbb R$ , $\nabla f(x)=0 \implies f $ is constant on $U$ ?
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12 views

General case for differentiation under the integral sign

What is the most convenient way to decide if we can differentiate under the integral sign? If the integrant is a smooth function, could we do so?
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1answer
33 views

Show that for each $a>0$ the function $e^{-ax}x^{a^2}$ has a maximum value, say $F(a)$, and that $F(x)$ has a minimum value $e^{-e/2}$

Show that for each $a>0$ the function $e^{-ax}x^{a^2}$ has a maximum value say $F(a)$,and that $F(x)$ has a minimum value $e^{-e/2}$. I differentiated the function $f(x)=e^{-ax}x^{a^2}$ to get ...
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3answers
33 views

The value of $a$ for which $f(x)=x^3+3(a-7)x^2+3(a^2-9)x-1$ have a positive point of maximum lies in the interval $(a_1,a_2)\cup(a_3,a_4)$

The value of $a$ for which $f(x)=x^3+3(a-7)x^2+3(a^2-9)x-1$ have a positive point of maximum lies in the interval $(a_1,a_2)\cup(a_3,a_4)$.find the value of $a_2+11a_3+70a_4$ I differentiated ...
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1answer
43 views

Proof of Green's identity

Can anyone explain to me how to prove Green's identity by integrating the divergence theorem? I don't understand how divergence, total derivative, and Laplace are related to each other. Why is this ...
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1answer
31 views

What is the radius of smallest circular disk large enough to cover every acute isosceles triangle of a given perimeter $L$?

What is the radius of smallest circular disk large enough to cover every acute isosceles triangle of a given perimeter $L$? Let $a,a,b$ are the sides of the isosceles triangle whose perimeter is ...
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1answer
38 views

Find the cosine of the angle at the vertex of an isosceles triangle having the greatest area

Find the cosine of the angle at the vertex of an isosceles triangle having the greatest area for the given constant length $l$ of the median drawn to its lateral side. I tried to solve this ...
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1answer
23 views

Find the values of the derivatives of the integral with a variable inside its limits.

$\require{cancel}$ Problem: I have the function $g: \mathbb{R} \to \mathbb{R}$ defined as $$ g(x)=\int^{(1+x^2)}_{-(1+x^2)} sin(t^3)\ dt,\ x \in \mathbb{R} $$ I would like to calculate values of ...
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1answer
28 views

If $f\in C^2(\mathbb R)$ then $M_1^2 \le 2M_0 M_2$, where $M_k = \text {sup}_x |(d/dx)^k f(x)|$ for $k=0,1,2.$

I wanna prove this problem. I tried it with Mean Value Theorem but cannot proceed to any plausible result. So could I have some hints?
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0answers
27 views

Find the maximum volume of the cylinder.

A cylinder is obtained by revolving a rectangle about the $x-$axis,the base of the rectangle lying on the $x-$axis and the entire rectangle lying in the region between the curve $y=\frac{x}{x^2+1}$ ...
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0answers
37 views

Questioning the differentiability of $f(x,y)$

$$f(x,y)=\begin{cases} y- \frac{e^{x^2+y^2}-x^2-y^2}{x^2+y^2},& x^2+y^2 \neq 0. \\ -1, & x=y=0 \end{cases}$$ I keep runnung into trouble with these types of questions. The way I do them is ...
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2answers
40 views

Let $p,q$ be real polynomials. Let $F: \Bbb R \to \Bbb R$ be differentiable, then $p=q$.

Let $p,q$ be real polynomials. Let $F: \Bbb R \to \Bbb R$ be differentiable, defined by: $$ F = \begin{cases} \hfill q \hfill & \text{X $\geq$ a} \\ \hfill p \hfill & ...
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3answers
29 views

Prove that the following functions is differentiable on $(-1,1) \times \mathbb R$

$$f(x,y)=\begin{cases} \frac{\tan x}{x}+y, & 0<|x|<1 \\ 1+y,& x=0 \\ \end{cases}$$ Prove that it is differentiable on $(-1,1) \times \mathbb R$. I use the Frechet definition of ...
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2answers
35 views

Differentiation using Chain Rule

Find $\frac{dy}{dx}$ if $y=7+5^{x^2+2x-1}$. So far I have done $\frac{dy}{dx}=(5^{x^2+2x-1})'$. Now, the RHS can be found by $(e^{\ln 5\cdot (x^2+2x-1)})'=e^{\ln 5\cdot (x^2+2x-1)}(x^2+2x-1)'\ln ...