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

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1
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0answers
12 views

Inequality with derivative and supremum norm

I have the following property written in a book but I can't understand why this implication is true. I would be glad if anyone could help me let $A \in \mathbb{R}^N$. $$\frac{d}{dt} \nabla A(x,t) = ...
2
votes
3answers
97 views

Applying the chain rule to compute $\frac{d}{dx}(\cos^6 x)$

$$\frac{d}{dx}(\cos^6x)$$ Using the chain rule $ M'(N(x)).N'(x)$, I'm deconstructing the $\cos$ function $$\begin{align*} &M= \cos^6 \\ &N= x\end{align*}$$ End result should be ...
1
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0answers
27 views

Differentiation of an integral in regards to different variables

It is known by the second fundamental that $$\frac{d}{dx}\int_0^x{\sin{(a \cdot t)}\ dt}=\sin{(a \cdot x)}$$ But what can we say about $$\frac{d}{da}\int_0^x{\sin{(a \cdot t)}\ dt}=\ ?$$
5
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2answers
88 views

Derivation of Dirac delta function

Is there anyone could give me a hint how to find the distributional derivative of the delta function $\delta$? I don't know how to deal with the infinite point.
1
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1answer
53 views

Find the derivative of $\left(\frac{4x+2}{x-2}\right)^5$

Hey helpful people I have one more question I am stuck on! $$f(x) = \left(\frac{4x+2}{x-2}\right)^5$$ I know the answer is $$\frac{-50(4x+2)^4}{(x-2)^2(x-2)^4}$$ But I really can't figure out how ...
1
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3answers
50 views

Find the derivative of $\frac{3}{x} - \frac{x}{2}$

I must find the derivative for: $\frac{3}{x} - \frac{x}{2}$ I know the answer is$ \frac{-3}{x^2} - \frac{1}{2}$ But I can't figure out why the 3 is negative and where the 1/2 came from Any help ...
1
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0answers
31 views

How does the step in the picture transition to step 2?

:) I have a math question regarding this picture. The problem is that I do not understand how the first equation turns into the the second. Where did the integral come from?? (the dv and dt) Update: ...
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votes
4answers
37 views

Find particular solution to nonhomogeneous DE $y'+y=x^2+\sin{x}+\cos{x}$

I'm new to nonhomogeneous DE's and I have come across this DE: $$y'+y=x^2+\sin{x}+\cos{x}$$ which I'm supposed to provide a general solution to. However, I get stuck with the particular solution. The ...
0
votes
0answers
32 views

$f$ is $3-$times differentiable and has at least $5$ distinct real zeroes, prove $f+6f'+12f''+8f'''$ has at least two distinct real zeroes?

Let $f$ be a three times differentiable function (defined on $\mathbb{R}$ and real-valued) such that $f$ has at least five distinct real zeroes. Prove that $f+6f'+12f''+8f'''$ has at least two ...
0
votes
0answers
23 views

Derivative atan2 of a function

I am not able to understand how to solve my doubt. I need to do the : $\frac{\partial}{\partial p} atan2({\cos(\alpha)},{\sin(\alpha)})$ I will compute $\cos(\alpha)$ and $\sin(\alpha)$ as: ...
0
votes
1answer
23 views

What is the full width of a peak of the function $F(X)=\frac{1+\cos((2N+1)πX)}{1+\cos(πX)}$

With $$1 + \cos \theta = 2 \cos^2 \frac{\theta}{2},$$ the function becomes $$f_n(x) = \left( \frac{\cos \frac{(2n+1)\pi x}{2}}{\cos \frac{\pi x}{2}} \right)^2.$$ It peaks at odd X integer values. ...
4
votes
1answer
31 views

Derivative of x|x| at 0

I am trying to show that $f(x) = x|x|$ is differentiable for all $x \in \mathbb{R}$. By computing the prime derivative I get: $$f'(x) = |x|+x(|x|)'$$ I know that $(|x|)' = \begin{cases} 1 \ ...
0
votes
1answer
39 views

$n$th derivative of function $\frac{1}{(1-2x)^2}$

I am trying to find the $n$th derivative of the function $\frac{1}{(1-2x)^2}$. The first three are simple but I can't see a schema right now. \begin{align*} y^{\prime} & = \frac{4}{(1-2x)^3}\\ ...
0
votes
1answer
23 views

Determine the Fourier series considering the derivative of a function

Let $f\left(x\right)=x^2+1$ on the interval $\left[-\pi,\pi\right]$, which is extended periodically to $\mathbb{R}$. I have calculated the Fourier series of $f$ to be ...
0
votes
2answers
46 views

Question on Rolle's theorem involving roots

Use Rolle's theorem to show that $f(x)=x^3-\frac{3}{2}x^2+\lambda$, $\lambda \in \mathbf{R}$ never has 2 zeroes in $[0,1]$. I started by assuming that $\exists$ $2$ zeroes in$[0,1]$ Then ...
2
votes
3answers
101 views

Real Analysis question on FTC, Integral

Let $g:[0,1] \rightarrow \mathbb R$ be a continuous function and assume that $$ \int_{0}^{1} g(x) \phi'(x) dx = 0 $$ for all continuously differentiable functions $\phi: [0,1] \rightarrow ...
0
votes
0answers
31 views

Problem finding the tangent plane and the normal line of an surface [closed]

Good night, I have a serious problem when I try to find a tangent plane for the following surface at the point $P$: $$x^{2}+y^{2}+z^{2}=6, \hspace{4mm} P=(-1,-2,3).$$ I make this: $\nabla ...
2
votes
2answers
50 views

How to find the set of values $S$ where $f$ is not differentiable?

Let's assume we are given an arbitrary function $f : \mathbb{R} \to \mathbb{R}$, and for the purposes of this question, let's assume we know nothing about the differentiability of $f$, i.e. we have no ...
3
votes
2answers
32 views

Relative Extrema - First-derivative test of : $f(x)=x^5-5x^3-20x-2$

Find the relative extrema of the function by applying the first-derivative test: $$f(x)=x^5-5x^3-20x-2$$ So I found the $f'(x)$ $$f'(x) = 5x^4-15x^2-20$$ Now, I'm trying to find the critical ...
5
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3answers
101 views

Derivative of $\log |AA^T|$ with respect to $A$.

What is the derivative of $\log |AA^T|$ with respect to $A$ where $|A|$ denotes the determinant of matrix A?
0
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1answer
24 views

Factoring $R(ry')'-y(rR')'=[r(Ry'-R'y)]'$

In a problem this formula was used and I'm not seeing how this factor using the chain rule was derived. Other than calculating the derivative of the two that someone else already solved and showing ...
2
votes
2answers
37 views

Partial Derivative of $xy^2+yz^2+xyz+x^2y^2z^2=5$

Someone can tell me what the Partial Derivative of $\frac{d^2z}{dy^2}$ of function $z(x,y)$ if it`s look like this: $$xy^2+yz^2+xyz+x^2y^2z^2=5$$ I try to solve the first derivative: ...
0
votes
1answer
41 views

If $f\colon\mathbb{R}\to\mathbb{R}$ satisfies $\lvert f(x)\rvert\le x^2$ for every $x\in\mathbb{R}$, then $f$ is differentiable at 0.

If $ f\colon \mathbb{R} \to \mathbb{R}$ satisfies $\lvert f(x)\rvert\le x^2 $ for every $x \in \mathbb{R} $, then $f$ is differentiable at $0$. The solution provided uses delta-epsilon to prove ...
3
votes
2answers
64 views

When is $\frac{dx}{dt}=\frac{\Delta x}{\Delta t}$ a valid approximation?

It is often said that when the change in e.g. $\Delta x$ is small than we can make the approximation: $$\frac{dx}{dt}=\frac{\Delta x}{\Delta t}$$ But it is not enough to say $\Delta x$ is small ...
0
votes
1answer
41 views

How the derivatives are different if sign changes.

I have this expression $$\frac{1}{(1 - x) ^ 2}$$ I need the derivative of this expression. So I calculated it, no big deal. However something has crossed my mind. Mathematically $(1 - x) ^ 2 = (x - 1) ...
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1answer
40 views

To check differentiability of function [closed]

A function $f: [0,3]\rightarrow \Bbb{R}$ is defined by $$f(x) = |x| + |x-1| + |x-2| + |x-3|\quad \forall x \in [0,3]$$ The number of points in $[0,3]$ where $f$ is not differentiable is
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3answers
37 views

$\frac{d}{d\theta}\mu=?$, where $\theta=\log\frac{\mu}{1-\mu}$ [closed]

How can I solve the following differentiation : $$\frac{d}{d\theta}\mu,$$ where $\theta=\log\frac{\mu}{1-\mu}$ ?
1
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1answer
48 views

Differentiation Involving Determinant.

I have to compute the following differentiation : $$\frac{\partial}{\partial\sigma^2}\det[\mathbf X_{p\times n}'(\sigma^2 \mathbf I_{n}+\mathbf Z_{n\times q}\mathbf G_{q\times q}\mathbf Z_{q\times ...
0
votes
0answers
15 views

Monotony and convexity of $U(t) = w(t) - t. w'(t)$

Let $\mathcal{D} = (\mathbb{R}^{+*})^2$. $c \in ]0,1[$. Moreover we have $\theta < 1$ and $\theta \ne 0$. We consider the following function (which is called CES or constant elasticity of ...
0
votes
0answers
9 views

Translating Logistic Regression loss function to Softmax

I currently have a program which takes a feature vector and classification, and applies it to a known weight vector to generate a loss gradient using Logistic Regression. This is that code: ...
2
votes
1answer
27 views

Mean-value Theorem $f(x)=\sqrt{x+2}; [4,6]$

Verify that the hypothesis of the mean-value theorem is satisfied for the given function on the indicated interval. Then find a suitable value for $c$ that satisfies the conclusion of the ...
0
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1answer
10 views

Finding the Components of a Hessian Matrix of a Quadratic Form

I'm trying to find the Hessian form of the following quadratic form: $f(x,y) = x^2y+y^2+xy$. I know that it's in the form of a matrix and that the elements of $H_f(a)_{i,j}=\dfrac{\delta^2f}{\delta ...
0
votes
2answers
41 views

Differentiate a Function (Help me Solve?!)

Find $\dfrac{d}{dx}$ for: $C(1+Ae^{-bt})^{-1}$ I have tried and arrived at: $-C(1+Ae^{-bt})^{-2}$ however that is not the correct answer.
0
votes
2answers
42 views

Extrema Where the Derivative is Undefined

Say we are given the derivative of a function say, $$f'(x)=\begin{cases} 5 & x<3 \\ -5 & x>3 \end{cases}$$ Notice that the derivative has opposite signs on either side of $x=3$, so you ...
5
votes
5answers
154 views

Finding the value of x at which the tangent to the curve is parallel to the x axis

I have thoroughly searched up how to attempt this question. However, I am not sure if my answer is correct or if I even attempted the question correctly. Assistance would be greatly appreciated! ...
1
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1answer
35 views

Prove Two Functions are Simultaneously Continuous

Let $f,g,h: \mathbb{R} \rightarrow \mathbb{R}$ so that $f$ is differentiable, $g,h$ monotone and $f'=f+g+h$. Prove that $g$ is continuous in $x_0$ iff $h$ continuous in $x_0$. My ...
4
votes
1answer
97 views

On an injective ring homomorphism from the ring of continuous functions to the ring of differentiable functions

Let $\phi : C \to D$ be an injective ring homomorphism such that $\phi(1)=1$, where $1$ denotes the constant function $1$ and $C,D$ are the rings of continuous and differentiable functions on ...
1
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2answers
79 views

Why does $\lim_{h \to 0^-} \frac{f(x+h) - f(x)}{h} \neq \lim_{h \to 0} \frac{f(x-h) - f(x)}{h} $

I realize that the only reason one-sided limits arise is as a result of the $\epsilon-\delta$ definition of a limit, applied to the real field $\mathbb{R}$, and that one-sided limits aren't even well ...
4
votes
4answers
79 views

Given $f(x) = x + |x|$ for what values of $x$ is $f$ differentiable

Problem : Given $f(x) = x + |x|$ for what values of $x$ is $f$ differentiable? For the sake of generality, let's assume that it is unknown to us that $|x|$ is not differentiable at $x = 0$ ...
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votes
1answer
43 views

Proving a function $f$ is not differentiable at an unkown point $a$

Let's say I have an arbitrary function $f : \mathbb{R} \to \mathbb{R}$, and I want to prove that it is not differentiable at some unknown point $a$. Emphasis must be placed on the unknown part as that ...
0
votes
1answer
42 views

Elastic curves - What is wrong about my solution?

Given a curve $\gamma:\mathbb R\to\mathbb R^2$ with $\Vert\gamma'\Vert=1$ and curvature $\kappa(s)=\frac{c}{\cosh s}$, $c\in\mathbb R$, how can I show that $\gamma$ is an elastic curve for some ...
0
votes
1answer
50 views

Alternate definition of differentability at a point

Usually in most introductory Calculus courses, a definition of differentiability at a point $a$ is defined, as follows : A function $f$ is differentiable at $a$ if $f'(a)$ exists As a corollary ...
0
votes
0answers
20 views

Determine differentiability in region

Is there any test to indirectly determine if a function f is differentiable everywhere in a defined domain, for example $S=\{x|x\in\mathbb{R}\land x^2<4\}$? For example: let $f(x)=\frac{2}{x}$. We ...
2
votes
1answer
56 views

Where did I go wrong in finding maximum?

The question in my book is given as: If $x^2+y^2+z^2=1$ for $x,y,z$ belongs to all real numbers ($x,y,z$ are independent), then find the maximum of $x^3+y^3+z^3-3xyz$. What I tried: As all ...
0
votes
1answer
16 views

Left and right derivative of composition

Let $f(x)=y$ and $g(y)$ be two functions such tat $f'_+(x)$(right sided derivative exists) and $g'_+(x)$ (right sided derivative exists) let $h(x)$ be a function that is defined as $h(x)=g(f(x))$ is ...
4
votes
2answers
272 views

Is there any solution to find a condition for $f(x)=a+bx^n+cx^2-dx>0$ to always hold true?

Okay, I am interested to know the criteria for a function to always hold $$f(x)=a+bx^n+cx^2-dx>0,$$ if it is given that $a, b, c>0$ and $n\in(-2,2)$ is some real number and $x>0$. My idea ...
2
votes
1answer
23 views

Sum of differentiable functions.

True/False Question : suppose that $f+g$ is differentiable at point $x_0$ therefore $f$ and $g$ are differentiable at $x_0$ I think this statement is false and I got a counter example : ...
1
vote
1answer
33 views

Local Immersion Theorem in $\mathbb{R}^n$ proof

I am trying to prove the following: Let $U \subset \mathbb{R}^n$ be open and $f \in C^1(U;\mathbb{R}^m)$. Let $x^\star \in U, \ y^\star = f(x^\star)$. Suppose that $\mathrm{d}f(x^\star)$ ...
-1
votes
1answer
50 views

Derivative Optimization Problem [duplicate]

I need help with finding the area of the largest rectangle in an ellipse from $y^2 + (x^2)/4 = 1$. I got it to y = $\sqrt{ 1 - (x^2)/4}$ but then I don't really know what to do, please help.
1
vote
2answers
24 views

Meaning behind directional derivative

My task was to find the directional derivative of function: $$z = y^2 - \sin(xy)$$ at the point $(0, -1)$ in direction of vector $u = (-1, 10) $. The result I found was $-21/\sqrt{101}$. But I ...