Questions tagged [derivatives]

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

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Is there any meaning for $f(x)-f'(x)$?

I am looking for any special property or application for $f(x)-f'(x)$. Like if it represents something about the main function f. Thanks in advance!
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Which comes first derivative of force then vector it or derivative of vector force.

I am trying to calculate $\vec{F}_{ij}$ where $F=\frac{1}{r^2}$ and $r$ is in $(x,y,z)$ coordinates that are function of $t$ for example $x(t)$. Then force in vector form is, $\vec{F}_{ij}=\frac{\...
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Derivative with respect to ct in schwarzschild metric

I was studying Christoffel symbols but came upon a question I wanted to ask. Let's say using the Schwarzschild metric, I wanted to find the term $$\frac{d Γ^0_{01}}{dx^0}$$ I know that $x^0=ct$ , and ...
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If a function is a constant a.e. on $[a,b]$, does it have the derivative a.e. on $[a,b]$ which equals to 0?

If a function is a constant a.e. on $[a,b]$, does it have the derivative a.e. on $[a,b]$ which equals to $0$?
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Solution of double derivative equations

In my book this question $xy''-3y'=4x^2$ is under the topic Linear & Bernoulli's equation. How do I solve this using the concept of $\frac{dy}{dx}+Py=Q$
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How to determine constants when taking the derivative?

I was doing my pre-exam revision and I seem to have gotten confused by thinking too much. I have the following derivative: $\dfrac d{dx}(xy)$. Now the my textbook tells me to use the product rule, but ...
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2 answers
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For all $\epsilon >0 $, exists $l(\epsilon)$ such that $\forall x \in \mathbb{R}$ and $\phi \in (0, l(\epsilon)), |f(x+\phi) -f(x)| < \epsilon$

Problem Let $f:\mathbb{R} \rightarrow \mathbb{R}$ differentiable such that $|f'(x)|<K$. Prove for all $\epsilon >0 $, exists $l(\epsilon)$ such that $\forall x \in \mathbb{R}$ and $\phi \in (0, ...
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Suggestions to evaluate a summation involving Hermite polynomials

I started with the the following four-variable function, $f(s,x,y,u)$, expressed as a summation involving the product of Hermite polynomials. $f(s,x,y,u)=\sum_{n=0}^{\infty}\frac{H_{n}(x)H_{n}(y)}{(n-...
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Why is it said that Laplacians "smooth solutions"?

Say you have some PDE for a function $f:[0,T]\times \mathbb{R}^n\to\mathbb C$ $$\frac{\partial f}{\partial t}=Lf,$$ for some differential operator $L$. If $L$ contains a Laplacian term $$\Delta f=\...
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an infinitely differentiable monotonic function that goes to 0 at infinity but f' does not

We want infinitely differentiable monotonic function f such that: $\lim_{x \to +\infty} f(x) = 0$ and $\lim_{x \to +\infty} f'(x) \neq 0$. An infinitely differentiable bridging function, strictly ...
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Finding the subgradient equations for sparse group lasso

In the SGL paper, the subgradient equations are shown in terms of group $k$, given by $$\frac{1}{n}X^{(k)\top} \left(y-\sum_{l=1}^m X^{(l)} \hat{\beta}^{(l)} \right)= (1-\alpha)\lambda u + \alpha\...
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Generalized Directional Derivative of $-|x| $ (minus absolute $x$) (Clarke Derivative)

I am having hard time calculating the Generalized Directional Derivative (Clarke Derivative) of $f(x)=-|x|$ at $x=0$. The answer is $f^{\circ}(0;v)=|v|$. The Generalized Directional Derivative is ...
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How to comput Hessian or second derivative using chain rule.

Suppose I have function $L=f(\mathbf{W}\mathbf{x})$, where $\mathbf{W}$ is a matrix, $\mathbf{x}$ is a vector, and the $f(\cdot)$ produces a scalar. I am wondering how to compute the Hessian of $L$ ...
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Find all $f:\mathbb{R}\to \mathbb{R}, f\in C^1$ so that $q, f(q)$ has the same denominator as $q$

Find all continuously differentiable functions $f:\mathbb{R}\to \mathbb{R}$ so that for every rational number $q, f(q)$ is rational and has the same denominator as $q$ in lowest terms (to simplify, ...
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Prove that $0<\dfrac{1}{x}\ln(\dfrac{e^x-1}{x})<1$ for $x>0$

I have the following question before me: Prove that $0<\dfrac{1}{x}\ln(\dfrac{e^x-1}{x})<1$ for $x>0$ using mean value theorem. I took $f(x)=\ln(\dfrac{e^x-1}{x})$ and applied LMVT on $f(x)$ ...
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Fourth Order Homogeneous Ordinary Differential Equation [closed]

Finding the homogenous solution for Y'''' - B^4Y = 0. The roots I found are ±B and ±Bi, so the solution I got was Y=C1eBx+C2 e-Bx+C3 sin(Bx)+C4 cos(Bx) I got the same result when using Wolfram Alpha. ...
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How to differentiate this equation? [closed]

I'm trying to differentiate the equation on the top to get the final answer on the bottom (both shown in this picture) as part of my maths classwork. However, I can't seem to figure out the RHS. I've ...
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Let $f(x)=2\arccos x+4\operatorname{ arccot } x-3x^2-2x+10, x\in[-1,1]$. If $[a,b]$ is the range of $f(x)$, find $4a-b$.

Question: Let $f(x)=2\arccos x+4\operatorname{ arccot } x-3x^2-2x+10, x\in[-1,1]$. If $[a,b]$ is the range of $f(x)$, find $4a-b$. Method $1:$ $f'(x)=-\frac{2}{\sqrt{1-x^2}}-\frac{4}{1+x^2}-6x-2$ 1st, ...
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Does Left Hand Derivate and Right Hand Derivative being defined guarantee continuity?

Suppose at $x = a$, both the Left Hand Derivative and Right Hand Derivative of a function exists and is defined. In other words, both the limits $$\lim_{h \to 0^-} \frac{f(a+h)-f(a)}{h}$$ and $$\lim_{...
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Can we take individual derivative of piecewise function if the function is continuous and differentiable?

Can we take individual derivative of piecewise function if the function is continuous and differentiable? Suppose a function $f(x)$ is defined in such a way that it's definition changes at some ...
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How do you explain calculation of rates (Multiplication vs Addition) to a non math student?

I had to explained Michaelis–Menten kinetics to a biology majoring students who had never taken advanced mathematics. During the calculation of rates I had to convince them we have to use the product (...
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1 vote
1 answer
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Complex differentiability and higher derivatives of given function

I have a question that goes: Is $f:\mathbb{C} \rightarrow \mathbb{C}$ given by $f(z)=z^2+z|z|^2$ differentiable at $z=0$? If yes, find $f'(0).$ Also, does $f^{(n)}(0)$ exist for $n \geq 2?$ So what I ...
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A closed-form for higher-order derivatives of the Dawson integral?

We have as the Dawson integral $$ \mathcal D(x):=e^{-x^2}\int_0^xe^{t^2}\,\mathrm dt. $$ I am interested in an expression for $\mathcal D^{(n)}(x)$ that does not involve sums. For example, we could ...
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2 answers
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Let $f:(0,1)\to \mathbb R$ be a twice differentiable function. Which of the following is/are FALSE?

Let $f:(0,1)\to \mathbb R$ be a twice differentiable function. Which of the following is/are FALSE? $a).$ If $f$ is bounded then $f'$ is bounded. $b).$ If $f$ and $f'$ are bounded then $f''$ is ...
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Confusion about Functional Derivative from Wikipedia

In the Wikipedia page and from its references, it states that given a functional $F:B \to \mathbb{R}$ (where B in my case is a Banach Space), its functional derivative is defined as $$ \frac{d}{d\...
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1 answer
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does there always exist a vector v such that, the partial derivative of a function $f:\mathbb{R}^n \to \mathbb{R}$ in the direction of v is zero?

Does there always exist a vector v such that, the partial derivative of a function $f:\mathbb{R}^n \to \mathbb{R}$ in the direction of v is zero? My approach would be saying that the directional ...
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Nested differential equation $y'(x)=y(y(x))$ [duplicate]

I like to solve a nested differential equation for example $$ y'(x)=y(y(x)),\tag 1 $$ where $x$ is real or in a real interval. A similar problem such (1) rises when I try to find special solutions to ...
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Differentiating expectation by a measure

Consider the following expression: $$ \frac{d\mathbb{E}_{\mu}[f(x)]}{d\mu} = \frac{d\int_{\mathcal{X}}f(x)d\mu}{d\mu} $$ where $x \in \mathcal{X}$ is the universal set associated with $\mu$. How do I ...
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Proof of boundedness involving gradients

This question relates to bounded second moment of stochastic noise in SGD. $\sigma^2$ represents the data variance and $\sigma_1^2$, $\sigma_2^2$ represents the data variance after partioning the data-...
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High-order complex derivative in MATLAB

First derivative can be calculated by the complex-step derivative formula: $f'(x)=\frac{Im(f(x+ih))}{h}$ Generalization of the above for calculating derivatives of any order employs multicomplex ...
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An equality concerning the differential quotient of $|z|$

I was trying to proof $|z|$ was not differentiable at any $z\in \mathbb{C} $. I gave it couple of tries but couldn't simplify the following equation $ \frac{|z+h| - |z|}{h} $. I searched online for ...
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Spivak: Is there an actual relationship between $dx$ and $du$ when we use a common shortcut to use the substitution formula for integration?

My question is at the very end of this post, in the final three paragraphs. I want to first lay out the context for the question, so that when it is stated perhaps it is a little clearer why I am ...
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2 votes
3 answers
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For what values ​of $k$, $f(x)=\lvert x^{2}+(k-1) \lvert x \rvert -k \rvert$ is non-differentiable at five points?

I arrived at the following answer by drawing graphs with a calculator: $$k<0 \quad \& \quad k≠-1$$ But I was hoping that maybe it would be possible to find the answer by using an algebraic ...
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0 answers
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how can i show which direction derivatives of a given function do exist? [closed]

i have a given function: f(x,y) = (0, if (x,y)=(0,0) or sin(x³+y³)/(x²+y²), if (x,y) =/= (0,0)) how can i show which directional derivatives do exist? i tried to approach it with the directional ...
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To Check the continuity and differentiability of the given function.

Suppose $A:=\{x\in [0,1]: x=\frac{p}{2^q},p\in \mathbb Z, q\in \mathbb N\}$ and $B:=\mathbb Z[\sqrt 2]\bigcap[0,1]$. Let $f:[0,1]\to \mathbb R$ be a function defined by $$\begin{align} f(x) = \begin{...
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1 answer
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Intuitive understanding for $f(x)=f'(x)$ in $(a,b)$ when $f(a)=f(b)=0$

Question: Let $f(x)$ be a differentiable function and $f(a)=f(b)=0 \;(a\lt b)$ then in the interval $(a,b)$ $f(x)+f'(x)=0$ has at least one root. $f(x)-f'(x)=0$ has at least one root. $f(x)\times f'...
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7 votes
5 answers
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Find $f^{(n)}(1)$ on $f(x)=(1+\sqrt{x})^{2n+2}$

Find $f^{(n)}(1)$ on $f(x)=(1+\sqrt{x})^{2n+2}$ . Here is a solution by someone: \begin{align*} f(x)&=(1+\sqrt{x})^{2n+2}=\sum_{k=0}^{2n+2}\binom{2n+2}{k}x^{\frac{k}{2}}\\ &=\sum_{k=0}^{2n+2}\...
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4 votes
4 answers
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Velocity with respect to position [closed]

We are given an expression for an object's position with respect to time. To find its velocity at $t=0,$ can we put $t=0$ into the position then divide by zero? If not, why?
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Spivak, Ch. 15, Problem 32: Doubt on step in proof of specific case of Sturm Comparison Theorem.

The following problem is from Chapter 15 of Spivak's Calculus. It is a question with multiple items, but my question is about two relatively simple steps in part $(c)$. *32. Suppose that $\phi_1$ and ...
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Are directional derivatives of continuous functions continuous as well? [closed]

I wonder if all directional derivatives of continuous functions are continuous as well. Intuitively this must be the case, as continuity is a conclusion of differentiability.
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Proving $\sum^n_{k=0} k^2 {n\choose k} = (n+n^2)2^{n-2}$

We can start with the expression of the binomial expansion, $\sum^n_{k=0} {n\choose k} x^ky^{n-k}= (x+y)^n$. Setting $x=y=1$ gives $\sum^n_{k=0} {n\choose k} = 2^n$ Differentiating both sides with ...
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Nth Derivative of a Function of Two Functions

This is a question related to the range-direction-cosine coordinate system, in case that helps. There is a unique property for this system in which one of the coordinate values, $w$, can be deduced ...
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Proving strong lyapunov function

I am working on the following problem: Let $x'=-ax+bf(y)$ and $y'=cx-df(y)$ with $f(0)=0$, $yf(y)>0$ for $y \neq 0$ and $a,b,c,d>0$. I need to show that the following function (for suitable ...
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2 answers
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Prove when Instaneous Velocity is equal to Average Velocity with Constant Acceleration

Assume constant acceleration. It seems that average velocity over some time interval [t1, t2], will be equal to the instantaneous velocity at the midpoint t = 1/2[t1 + t2]. I'm wondering how you might ...
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-3 votes
0 answers
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Finding equation of a curve given single point [closed]

I was recently sent an image of a question found in a book as a challange. But unfortunately I am unable to find the solution, without finding the equation of a curve. Below is the question. I just ...
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find $inf_C \int_0^1 |f'(x) - f(x)|dx$

Let $\mathbb{R}$ be the reals and $C$ the set of all functions $f:[0,1]\to \mathbb{R}$ with a continuous derivative and satisfying $f(0) = 0, f(1) = 1.$ Find $\inf_C \int_0^1 |f'(x)-f(x)|dx$. In the ...
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-1 votes
1 answer
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Solve a problem to minimise a cost using derivatives [closed]

Solving a problem to minimize cost A cylindrical chemical storage tank with a capacity of 1000 m3 is going to be constructed in a warehouse that is 14 m by 17 m, with a height of 11 m. The ...
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1 answer
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Meaning of $\lim_{h \to 0^+} \frac{g(2+h)-g(2-h)-16}{h}$

$g(x) = \begin{cases} f(x) & (x<a) \\ x^4 & (x \ge a) \end{cases}$ $f(x)$ is differentiable in every $x \in \mathbb{R}$. I was solving a math problem, and I've approached this formula : $$\...
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2 votes
0 answers
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Spivak, Ch. 15, Problem 31: Prove that $\sin$ isn't defined implicitly by an algebraic equation. Understanding a step in the proof.

The following is a problem from Ch. 15 of Spivak's Calculus (a) After all the work involved in the definition of $\sin$, it would be disconcerting to find that $\sin$ is actually a rational function. ...
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0 votes
1 answer
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Can I say $Y-$axis is a tangent to $f(x)$ at origin?

$f(x) = x^2$ when $x \geq 0$ I think any line passing through origin is a tangent to $f(x)$ at the origin. Please do not close the question. Can anyone answer me ?
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